| Step |
Hyp |
Ref |
Expression |
| 1 |
|
noetainflem.1 |
⊢ 𝑇 = if ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 , ( ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) , 1o ⟩ } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
| 2 |
|
noetainflem.2 |
⊢ 𝑊 = ( 𝑇 ∪ ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ) |
| 3 |
|
simplrl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ 𝑎 ∈ 𝐴 ) → 𝐵 ⊆ No ) |
| 4 |
|
simplrr |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ 𝑎 ∈ 𝐴 ) → 𝐵 ∈ V ) |
| 5 |
|
simpll |
⊢ ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) → 𝐴 ⊆ No ) |
| 6 |
5
|
sselda |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ∈ No ) |
| 7 |
1
|
noinfbnd2 |
⊢ ( ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ 𝑎 ∈ No ) → ( ∀ 𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) |
| 8 |
3 4 6 7
|
syl3anc |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ 𝑎 ∈ 𝐴 ) → ( ∀ 𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) |
| 9 |
|
simplll |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → 𝐴 ⊆ No ) |
| 10 |
|
simprl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → 𝑎 ∈ 𝐴 ) |
| 11 |
9 10
|
sseldd |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → 𝑎 ∈ No ) |
| 12 |
|
nodmord |
⊢ ( 𝑎 ∈ No → Ord dom 𝑎 ) |
| 13 |
|
ordirr |
⊢ ( Ord dom 𝑎 → ¬ dom 𝑎 ∈ dom 𝑎 ) |
| 14 |
11 12 13
|
3syl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → ¬ dom 𝑎 ∈ dom 𝑎 ) |
| 15 |
|
bdayval |
⊢ ( 𝑎 ∈ No → ( bday ‘ 𝑎 ) = dom 𝑎 ) |
| 16 |
11 15
|
syl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → ( bday ‘ 𝑎 ) = dom 𝑎 ) |
| 17 |
|
bdayfo |
⊢ bday : No –onto→ On |
| 18 |
|
fofn |
⊢ ( bday : No –onto→ On → bday Fn No ) |
| 19 |
17 18
|
ax-mp |
⊢ bday Fn No |
| 20 |
|
fnfvima |
⊢ ( ( bday Fn No ∧ 𝐴 ⊆ No ∧ 𝑎 ∈ 𝐴 ) → ( bday ‘ 𝑎 ) ∈ ( bday “ 𝐴 ) ) |
| 21 |
19 9 10 20
|
mp3an2i |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → ( bday ‘ 𝑎 ) ∈ ( bday “ 𝐴 ) ) |
| 22 |
16 21
|
eqeltrrd |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → dom 𝑎 ∈ ( bday “ 𝐴 ) ) |
| 23 |
|
elssuni |
⊢ ( dom 𝑎 ∈ ( bday “ 𝐴 ) → dom 𝑎 ⊆ ∪ ( bday “ 𝐴 ) ) |
| 24 |
22 23
|
syl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → dom 𝑎 ⊆ ∪ ( bday “ 𝐴 ) ) |
| 25 |
|
nodmon |
⊢ ( 𝑎 ∈ No → dom 𝑎 ∈ On ) |
| 26 |
|
imassrn |
⊢ ( bday “ 𝐴 ) ⊆ ran bday |
| 27 |
|
forn |
⊢ ( bday : No –onto→ On → ran bday = On ) |
| 28 |
17 27
|
ax-mp |
⊢ ran bday = On |
| 29 |
26 28
|
sseqtri |
⊢ ( bday “ 𝐴 ) ⊆ On |
| 30 |
|
ssorduni |
⊢ ( ( bday “ 𝐴 ) ⊆ On → Ord ∪ ( bday “ 𝐴 ) ) |
| 31 |
29 30
|
ax-mp |
⊢ Ord ∪ ( bday “ 𝐴 ) |
| 32 |
|
ordsssuc |
⊢ ( ( dom 𝑎 ∈ On ∧ Ord ∪ ( bday “ 𝐴 ) ) → ( dom 𝑎 ⊆ ∪ ( bday “ 𝐴 ) ↔ dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) ) ) |
| 33 |
31 32
|
mpan2 |
⊢ ( dom 𝑎 ∈ On → ( dom 𝑎 ⊆ ∪ ( bday “ 𝐴 ) ↔ dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) ) ) |
| 34 |
11 25 33
|
3syl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → ( dom 𝑎 ⊆ ∪ ( bday “ 𝐴 ) ↔ dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) ) ) |
| 35 |
24 34
|
mpbid |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) ) |
| 36 |
|
elun2 |
⊢ ( dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) → dom 𝑎 ∈ ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) ) |
| 37 |
35 36
|
syl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → dom 𝑎 ∈ ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) ) |
| 38 |
|
eleq2 |
⊢ ( dom 𝑎 = ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) → ( dom 𝑎 ∈ dom 𝑎 ↔ dom 𝑎 ∈ ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) ) ) |
| 39 |
37 38
|
syl5ibrcom |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → ( dom 𝑎 = ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) → dom 𝑎 ∈ dom 𝑎 ) ) |
| 40 |
14 39
|
mtod |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → ¬ dom 𝑎 = ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) ) |
| 41 |
|
dmeq |
⊢ ( 𝑎 = 𝑊 → dom 𝑎 = dom 𝑊 ) |
| 42 |
2
|
dmeqi |
⊢ dom 𝑊 = dom ( 𝑇 ∪ ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ) |
| 43 |
|
dmun |
⊢ dom ( 𝑇 ∪ ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ) = ( dom 𝑇 ∪ dom ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ) |
| 44 |
|
2oex |
⊢ 2o ∈ V |
| 45 |
44
|
snnz |
⊢ { 2o } ≠ ∅ |
| 46 |
|
dmxp |
⊢ ( { 2o } ≠ ∅ → dom ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) = ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) ) |
| 47 |
45 46
|
ax-mp |
⊢ dom ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) = ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) |
| 48 |
47
|
uneq2i |
⊢ ( dom 𝑇 ∪ dom ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ) = ( dom 𝑇 ∪ ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) ) |
| 49 |
|
undif2 |
⊢ ( dom 𝑇 ∪ ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) ) = ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) |
| 50 |
48 49
|
eqtri |
⊢ ( dom 𝑇 ∪ dom ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ) = ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) |
| 51 |
42 43 50
|
3eqtri |
⊢ dom 𝑊 = ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) |
| 52 |
41 51
|
eqtrdi |
⊢ ( 𝑎 = 𝑊 → dom 𝑎 = ( dom 𝑇 ∪ suc ∪ ( bday “ 𝐴 ) ) ) |
| 53 |
40 52
|
nsyl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → ¬ 𝑎 = 𝑊 ) |
| 54 |
53
|
neqned |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → 𝑎 ≠ 𝑊 ) |
| 55 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) |
| 56 |
11
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → 𝑎 ∈ No ) |
| 57 |
56
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → 𝑎 ∈ No ) |
| 58 |
|
simp-4r |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → 𝐴 ∈ V ) |
| 59 |
|
simplrl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → 𝐵 ⊆ No ) |
| 60 |
59
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → 𝐵 ⊆ No ) |
| 61 |
|
simplrr |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → 𝐵 ∈ V ) |
| 62 |
61
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → 𝐵 ∈ V ) |
| 63 |
1 2
|
noetainflem1 |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ⊆ No ∧ 𝐵 ∈ V ) → 𝑊 ∈ No ) |
| 64 |
58 60 62 63
|
syl3anc |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → 𝑊 ∈ No ) |
| 65 |
64
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → 𝑊 ∈ No ) |
| 66 |
|
simplr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → 𝑎 ≠ 𝑊 ) |
| 67 |
|
nosepne |
⊢ ( ( 𝑎 ∈ No ∧ 𝑊 ∈ No ∧ 𝑎 ≠ 𝑊 ) → ( 𝑎 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ≠ ( 𝑊 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 68 |
57 65 66 67
|
syl3anc |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑎 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ≠ ( 𝑊 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 69 |
55
|
fvresd |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( ( 𝑊 ↾ dom 𝑇 ) ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = ( 𝑊 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 70 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) |
| 71 |
1 2
|
noetainflem2 |
⊢ ( ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) → ( 𝑊 ↾ dom 𝑇 ) = 𝑇 ) |
| 72 |
70 71
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑊 ↾ dom 𝑇 ) = 𝑇 ) |
| 73 |
72
|
fveq1d |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( ( 𝑊 ↾ dom 𝑇 ) ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = ( 𝑇 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 74 |
69 73
|
eqtr3d |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑊 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = ( 𝑇 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 75 |
68 74
|
neeqtrd |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑎 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ≠ ( 𝑇 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 76 |
75
|
necomd |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑇 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ≠ ( 𝑎 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 77 |
|
fveq2 |
⊢ ( 𝑞 = ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } → ( 𝑇 ‘ 𝑞 ) = ( 𝑇 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 78 |
|
fveq2 |
⊢ ( 𝑞 = ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } → ( 𝑎 ‘ 𝑞 ) = ( 𝑎 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 79 |
77 78
|
neeq12d |
⊢ ( 𝑞 = ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } → ( ( 𝑇 ‘ 𝑞 ) ≠ ( 𝑎 ‘ 𝑞 ) ↔ ( 𝑇 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ≠ ( 𝑎 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) ) |
| 80 |
79
|
rspcev |
⊢ ( ( ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ∧ ( 𝑇 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ≠ ( 𝑎 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) → ∃ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) ≠ ( 𝑎 ‘ 𝑞 ) ) |
| 81 |
55 76 80
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ∃ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) ≠ ( 𝑎 ‘ 𝑞 ) ) |
| 82 |
|
df-ne |
⊢ ( ( 𝑇 ‘ 𝑞 ) ≠ ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ↔ ¬ ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) |
| 83 |
|
fvres |
⊢ ( 𝑞 ∈ dom 𝑇 → ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) = ( 𝑎 ‘ 𝑞 ) ) |
| 84 |
83
|
neeq2d |
⊢ ( 𝑞 ∈ dom 𝑇 → ( ( 𝑇 ‘ 𝑞 ) ≠ ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ↔ ( 𝑇 ‘ 𝑞 ) ≠ ( 𝑎 ‘ 𝑞 ) ) ) |
| 85 |
82 84
|
bitr3id |
⊢ ( 𝑞 ∈ dom 𝑇 → ( ¬ ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ↔ ( 𝑇 ‘ 𝑞 ) ≠ ( 𝑎 ‘ 𝑞 ) ) ) |
| 86 |
85
|
rexbiia |
⊢ ( ∃ 𝑞 ∈ dom 𝑇 ¬ ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ↔ ∃ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) ≠ ( 𝑎 ‘ 𝑞 ) ) |
| 87 |
|
rexnal |
⊢ ( ∃ 𝑞 ∈ dom 𝑇 ¬ ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ↔ ¬ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) |
| 88 |
86 87
|
bitr3i |
⊢ ( ∃ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) ≠ ( 𝑎 ‘ 𝑞 ) ↔ ¬ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) |
| 89 |
81 88
|
sylib |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ¬ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) |
| 90 |
89
|
olcd |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( ¬ dom 𝑇 = dom ( 𝑎 ↾ dom 𝑇 ) ∨ ¬ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) ) |
| 91 |
1
|
noinfno |
⊢ ( ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) → 𝑇 ∈ No ) |
| 92 |
91
|
ad3antlr |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → 𝑇 ∈ No ) |
| 93 |
92
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → 𝑇 ∈ No ) |
| 94 |
|
nofun |
⊢ ( 𝑇 ∈ No → Fun 𝑇 ) |
| 95 |
93 94
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → Fun 𝑇 ) |
| 96 |
|
nofun |
⊢ ( 𝑎 ∈ No → Fun 𝑎 ) |
| 97 |
|
funres |
⊢ ( Fun 𝑎 → Fun ( 𝑎 ↾ dom 𝑇 ) ) |
| 98 |
57 96 97
|
3syl |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → Fun ( 𝑎 ↾ dom 𝑇 ) ) |
| 99 |
|
eqfunfv |
⊢ ( ( Fun 𝑇 ∧ Fun ( 𝑎 ↾ dom 𝑇 ) ) → ( 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ↔ ( dom 𝑇 = dom ( 𝑎 ↾ dom 𝑇 ) ∧ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) ) ) |
| 100 |
95 98 99
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ↔ ( dom 𝑇 = dom ( 𝑎 ↾ dom 𝑇 ) ∧ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) ) ) |
| 101 |
|
ianor |
⊢ ( ¬ ( dom 𝑇 = dom ( 𝑎 ↾ dom 𝑇 ) ∧ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) ↔ ( ¬ dom 𝑇 = dom ( 𝑎 ↾ dom 𝑇 ) ∨ ¬ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) ) |
| 102 |
101
|
con1bii |
⊢ ( ¬ ( ¬ dom 𝑇 = dom ( 𝑎 ↾ dom 𝑇 ) ∨ ¬ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) ↔ ( dom 𝑇 = dom ( 𝑎 ↾ dom 𝑇 ) ∧ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) ) |
| 103 |
100 102
|
bitr4di |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ↔ ¬ ( ¬ dom 𝑇 = dom ( 𝑎 ↾ dom 𝑇 ) ∨ ¬ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) ) ) |
| 104 |
103
|
con2bid |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( ( ¬ dom 𝑇 = dom ( 𝑎 ↾ dom 𝑇 ) ∨ ¬ ∀ 𝑞 ∈ dom 𝑇 ( 𝑇 ‘ 𝑞 ) = ( ( 𝑎 ↾ dom 𝑇 ) ‘ 𝑞 ) ) ↔ ¬ 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ) ) |
| 105 |
90 104
|
mpbid |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ¬ 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ) |
| 106 |
105
|
pm2.21d |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑇 = ( 𝑎 ↾ dom 𝑇 ) → 𝑎 <s 𝑊 ) ) |
| 107 |
72
|
breq2d |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( ( 𝑎 ↾ dom 𝑇 ) <s ( 𝑊 ↾ dom 𝑇 ) ↔ ( 𝑎 ↾ dom 𝑇 ) <s 𝑇 ) ) |
| 108 |
|
nodmon |
⊢ ( 𝑇 ∈ No → dom 𝑇 ∈ On ) |
| 109 |
92 108
|
syl |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → dom 𝑇 ∈ On ) |
| 110 |
109
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → dom 𝑇 ∈ On ) |
| 111 |
|
ltsres |
⊢ ( ( 𝑎 ∈ No ∧ 𝑊 ∈ No ∧ dom 𝑇 ∈ On ) → ( ( 𝑎 ↾ dom 𝑇 ) <s ( 𝑊 ↾ dom 𝑇 ) → 𝑎 <s 𝑊 ) ) |
| 112 |
57 65 110 111
|
syl3anc |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( ( 𝑎 ↾ dom 𝑇 ) <s ( 𝑊 ↾ dom 𝑇 ) → 𝑎 <s 𝑊 ) ) |
| 113 |
107 112
|
sylbird |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( ( 𝑎 ↾ dom 𝑇 ) <s 𝑇 → 𝑎 <s 𝑊 ) ) |
| 114 |
|
simplrr |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) |
| 115 |
114
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) |
| 116 |
|
noreson |
⊢ ( ( 𝑎 ∈ No ∧ dom 𝑇 ∈ On ) → ( 𝑎 ↾ dom 𝑇 ) ∈ No ) |
| 117 |
56 109 116
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → ( 𝑎 ↾ dom 𝑇 ) ∈ No ) |
| 118 |
117
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑎 ↾ dom 𝑇 ) ∈ No ) |
| 119 |
|
ltsso |
⊢ <s Or No |
| 120 |
|
sotric |
⊢ ( ( <s Or No ∧ ( 𝑇 ∈ No ∧ ( 𝑎 ↾ dom 𝑇 ) ∈ No ) ) → ( 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ↔ ¬ ( 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ∨ ( 𝑎 ↾ dom 𝑇 ) <s 𝑇 ) ) ) |
| 121 |
119 120
|
mpan |
⊢ ( ( 𝑇 ∈ No ∧ ( 𝑎 ↾ dom 𝑇 ) ∈ No ) → ( 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ↔ ¬ ( 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ∨ ( 𝑎 ↾ dom 𝑇 ) <s 𝑇 ) ) ) |
| 122 |
93 118 121
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ↔ ¬ ( 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ∨ ( 𝑎 ↾ dom 𝑇 ) <s 𝑇 ) ) ) |
| 123 |
122
|
con2bid |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( ( 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ∨ ( 𝑎 ↾ dom 𝑇 ) <s 𝑇 ) ↔ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) |
| 124 |
115 123
|
mpbird |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → ( 𝑇 = ( 𝑎 ↾ dom 𝑇 ) ∨ ( 𝑎 ↾ dom 𝑇 ) <s 𝑇 ) ) |
| 125 |
106 113 124
|
mpjaod |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) → 𝑎 <s 𝑊 ) |
| 126 |
64
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → 𝑊 ∈ No ) |
| 127 |
56
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → 𝑎 ∈ No ) |
| 128 |
|
simplr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → 𝑎 ≠ 𝑊 ) |
| 129 |
128
|
necomd |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → 𝑊 ≠ 𝑎 ) |
| 130 |
2
|
fveq1i |
⊢ ( 𝑊 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = ( ( 𝑇 ∪ ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ) ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) |
| 131 |
92
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → 𝑇 ∈ No ) |
| 132 |
131 94
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → Fun 𝑇 ) |
| 133 |
132
|
funfnd |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → 𝑇 Fn dom 𝑇 ) |
| 134 |
|
fnconstg |
⊢ ( 2o ∈ V → ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) Fn ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) ) |
| 135 |
44 134
|
ax-mp |
⊢ ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) Fn ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) |
| 136 |
135
|
a1i |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) Fn ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) ) |
| 137 |
|
disjdif |
⊢ ( dom 𝑇 ∩ ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) ) = ∅ |
| 138 |
137
|
a1i |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( dom 𝑇 ∩ ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) ) = ∅ ) |
| 139 |
|
nosepssdm |
⊢ ( ( 𝑎 ∈ No ∧ 𝑊 ∈ No ∧ 𝑎 ≠ 𝑊 ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ⊆ dom 𝑎 ) |
| 140 |
127 126 128 139
|
syl3anc |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ⊆ dom 𝑎 ) |
| 141 |
127 15
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( bday ‘ 𝑎 ) = dom 𝑎 ) |
| 142 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → 𝐴 ⊆ No ) |
| 143 |
|
simplrl |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → 𝑎 ∈ 𝐴 ) |
| 144 |
143
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → 𝑎 ∈ 𝐴 ) |
| 145 |
19 142 144 20
|
mp3an2i |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( bday ‘ 𝑎 ) ∈ ( bday “ 𝐴 ) ) |
| 146 |
|
elssuni |
⊢ ( ( bday ‘ 𝑎 ) ∈ ( bday “ 𝐴 ) → ( bday ‘ 𝑎 ) ⊆ ∪ ( bday “ 𝐴 ) ) |
| 147 |
145 146
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( bday ‘ 𝑎 ) ⊆ ∪ ( bday “ 𝐴 ) ) |
| 148 |
141 147
|
eqsstrrd |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → dom 𝑎 ⊆ ∪ ( bday “ 𝐴 ) ) |
| 149 |
127 25 33
|
3syl |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( dom 𝑎 ⊆ ∪ ( bday “ 𝐴 ) ↔ dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) ) ) |
| 150 |
148 149
|
mpbid |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) ) |
| 151 |
|
simpr |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → 𝑎 ≠ 𝑊 ) |
| 152 |
|
nosepon |
⊢ ( ( 𝑎 ∈ No ∧ 𝑊 ∈ No ∧ 𝑎 ≠ 𝑊 ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ On ) |
| 153 |
56 64 151 152
|
syl3anc |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ On ) |
| 154 |
153
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ On ) |
| 155 |
|
eloni |
⊢ ( ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ On → Ord ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) |
| 156 |
|
ordsuc |
⊢ ( Ord ∪ ( bday “ 𝐴 ) ↔ Ord suc ∪ ( bday “ 𝐴 ) ) |
| 157 |
31 156
|
mpbi |
⊢ Ord suc ∪ ( bday “ 𝐴 ) |
| 158 |
|
ordtr2 |
⊢ ( ( Ord ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∧ Ord suc ∪ ( bday “ 𝐴 ) ) → ( ( ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ⊆ dom 𝑎 ∧ dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ suc ∪ ( bday “ 𝐴 ) ) ) |
| 159 |
157 158
|
mpan2 |
⊢ ( Ord ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } → ( ( ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ⊆ dom 𝑎 ∧ dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ suc ∪ ( bday “ 𝐴 ) ) ) |
| 160 |
154 155 159
|
3syl |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( ( ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ⊆ dom 𝑎 ∧ dom 𝑎 ∈ suc ∪ ( bday “ 𝐴 ) ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ suc ∪ ( bday “ 𝐴 ) ) ) |
| 161 |
140 150 160
|
mp2and |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ suc ∪ ( bday “ 𝐴 ) ) |
| 162 |
|
ontri1 |
⊢ ( ( dom 𝑇 ∈ On ∧ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ On ) → ( dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ↔ ¬ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) ) |
| 163 |
109 153 162
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → ( dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ↔ ¬ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) ) |
| 164 |
163
|
biimpa |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ¬ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ) |
| 165 |
161 164
|
eldifd |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) ) |
| 166 |
133 136 138 165
|
fvun2d |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( ( 𝑇 ∪ ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ) ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = ( ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 167 |
44
|
fvconst2 |
⊢ ( ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) → ( ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = 2o ) |
| 168 |
165 167
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = 2o ) |
| 169 |
166 168
|
eqtrd |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( ( 𝑇 ∪ ( ( suc ∪ ( bday “ 𝐴 ) ∖ dom 𝑇 ) × { 2o } ) ) ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = 2o ) |
| 170 |
130 169
|
eqtrid |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → ( 𝑊 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = 2o ) |
| 171 |
|
nosep2o |
⊢ ( ( ( 𝑊 ∈ No ∧ 𝑎 ∈ No ∧ 𝑊 ≠ 𝑎 ) ∧ ( 𝑊 ‘ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) = 2o ) → 𝑎 <s 𝑊 ) |
| 172 |
126 127 129 170 171
|
syl31anc |
⊢ ( ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) ∧ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) → 𝑎 <s 𝑊 ) |
| 173 |
153 155
|
syl |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → Ord ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) |
| 174 |
|
nodmord |
⊢ ( 𝑇 ∈ No → Ord dom 𝑇 ) |
| 175 |
92 174
|
syl |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → Ord dom 𝑇 ) |
| 176 |
|
ordtri2or |
⊢ ( ( Ord ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∧ Ord dom 𝑇 ) → ( ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ∨ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 177 |
173 175 176
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → ( ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ∈ dom 𝑇 ∨ dom 𝑇 ⊆ ∩ { 𝑝 ∈ On ∣ ( 𝑎 ‘ 𝑝 ) ≠ ( 𝑊 ‘ 𝑝 ) } ) ) |
| 178 |
125 172 177
|
mpjaodan |
⊢ ( ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) ∧ 𝑎 ≠ 𝑊 ) → 𝑎 <s 𝑊 ) |
| 179 |
54 178
|
mpdan |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ ( 𝑎 ∈ 𝐴 ∧ ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) ) ) → 𝑎 <s 𝑊 ) |
| 180 |
179
|
expr |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ 𝑎 ∈ 𝐴 ) → ( ¬ 𝑇 <s ( 𝑎 ↾ dom 𝑇 ) → 𝑎 <s 𝑊 ) ) |
| 181 |
8 180
|
sylbid |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) ∧ 𝑎 ∈ 𝐴 ) → ( ∀ 𝑏 ∈ 𝐵 𝑎 <s 𝑏 → 𝑎 <s 𝑊 ) ) |
| 182 |
181
|
ralimdva |
⊢ ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ) → ( ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 𝑎 <s 𝑏 → ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑊 ) ) |
| 183 |
182
|
3impia |
⊢ ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ V ) ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 𝑎 <s 𝑏 ) → ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑊 ) |