Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → 𝐵 ∈ FinIII ) |
2 |
|
simpll1 |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → 𝐴 ⊆ 𝒫 𝐵 ) |
3 |
2
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) ∧ 𝑒 ∈ ω ) → 𝐴 ⊆ 𝒫 𝐵 ) |
4 |
|
ssrab2 |
⊢ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ⊆ 𝐴 |
5 |
4
|
unissi |
⊢ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ⊆ ∪ 𝐴 |
6 |
|
sspwuni |
⊢ ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵 ) |
7 |
6
|
biimpi |
⊢ ( 𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ 𝐵 ) |
8 |
5 7
|
sstrid |
⊢ ( 𝐴 ⊆ 𝒫 𝐵 → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ⊆ 𝐵 ) |
9 |
3 8
|
syl |
⊢ ( ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) ∧ 𝑒 ∈ ω ) → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ⊆ 𝐵 ) |
10 |
|
elpw2g |
⊢ ( 𝐵 ∈ FinIII → ( ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ∈ 𝒫 𝐵 ↔ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ⊆ 𝐵 ) ) |
11 |
10
|
ad2antlr |
⊢ ( ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) ∧ 𝑒 ∈ ω ) → ( ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ∈ 𝒫 𝐵 ↔ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ⊆ 𝐵 ) ) |
12 |
9 11
|
mpbird |
⊢ ( ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) ∧ 𝑒 ∈ ω ) → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ∈ 𝒫 𝐵 ) |
13 |
12
|
fmpttd |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) : ω ⟶ 𝒫 𝐵 ) |
14 |
|
vex |
⊢ 𝑑 ∈ V |
15 |
14
|
sucex |
⊢ suc 𝑑 ∈ V |
16 |
|
sssucid |
⊢ 𝑑 ⊆ suc 𝑑 |
17 |
|
ssdomg |
⊢ ( suc 𝑑 ∈ V → ( 𝑑 ⊆ suc 𝑑 → 𝑑 ≼ suc 𝑑 ) ) |
18 |
15 16 17
|
mp2 |
⊢ 𝑑 ≼ suc 𝑑 |
19 |
|
domtr |
⊢ ( ( 𝑓 ≼ 𝑑 ∧ 𝑑 ≼ suc 𝑑 ) → 𝑓 ≼ suc 𝑑 ) |
20 |
18 19
|
mpan2 |
⊢ ( 𝑓 ≼ 𝑑 → 𝑓 ≼ suc 𝑑 ) |
21 |
20
|
a1i |
⊢ ( 𝑓 ∈ 𝐴 → ( 𝑓 ≼ 𝑑 → 𝑓 ≼ suc 𝑑 ) ) |
22 |
21
|
ss2rabi |
⊢ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ⊆ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } |
23 |
|
uniss |
⊢ ( { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ⊆ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ⊆ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ) |
24 |
22 23
|
mp1i |
⊢ ( ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) ∧ 𝑑 ∈ ω ) → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ⊆ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ) |
25 |
|
id |
⊢ ( 𝑑 ∈ ω → 𝑑 ∈ ω ) |
26 |
|
pwexg |
⊢ ( 𝐵 ∈ FinIII → 𝒫 𝐵 ∈ V ) |
27 |
26
|
adantl |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → 𝒫 𝐵 ∈ V ) |
28 |
27 2
|
ssexd |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → 𝐴 ∈ V ) |
29 |
|
rabexg |
⊢ ( 𝐴 ∈ V → { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ∈ V ) |
30 |
|
uniexg |
⊢ ( { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ∈ V → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ∈ V ) |
31 |
28 29 30
|
3syl |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ∈ V ) |
32 |
|
breq2 |
⊢ ( 𝑒 = 𝑑 → ( 𝑓 ≼ 𝑒 ↔ 𝑓 ≼ 𝑑 ) ) |
33 |
32
|
rabbidv |
⊢ ( 𝑒 = 𝑑 → { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } = { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ) |
34 |
33
|
unieqd |
⊢ ( 𝑒 = 𝑑 → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } = ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ) |
35 |
|
eqid |
⊢ ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) |
36 |
34 35
|
fvmptg |
⊢ ( ( 𝑑 ∈ ω ∧ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ∈ V ) → ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ 𝑑 ) = ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ) |
37 |
25 31 36
|
syl2anr |
⊢ ( ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) ∧ 𝑑 ∈ ω ) → ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ 𝑑 ) = ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑 } ) |
38 |
|
peano2 |
⊢ ( 𝑑 ∈ ω → suc 𝑑 ∈ ω ) |
39 |
|
rabexg |
⊢ ( 𝐴 ∈ V → { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ∈ V ) |
40 |
|
uniexg |
⊢ ( { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ∈ V → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ∈ V ) |
41 |
28 39 40
|
3syl |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ∈ V ) |
42 |
|
breq2 |
⊢ ( 𝑒 = suc 𝑑 → ( 𝑓 ≼ 𝑒 ↔ 𝑓 ≼ suc 𝑑 ) ) |
43 |
42
|
rabbidv |
⊢ ( 𝑒 = suc 𝑑 → { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } = { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ) |
44 |
43
|
unieqd |
⊢ ( 𝑒 = suc 𝑑 → ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } = ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ) |
45 |
44 35
|
fvmptg |
⊢ ( ( suc 𝑑 ∈ ω ∧ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ∈ V ) → ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ suc 𝑑 ) = ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ) |
46 |
38 41 45
|
syl2anr |
⊢ ( ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) ∧ 𝑑 ∈ ω ) → ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ suc 𝑑 ) = ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑 } ) |
47 |
24 37 46
|
3sstr4d |
⊢ ( ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) ∧ 𝑑 ∈ ω ) → ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ 𝑑 ) ⊆ ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ suc 𝑑 ) ) |
48 |
47
|
ralrimiva |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ∀ 𝑑 ∈ ω ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ 𝑑 ) ⊆ ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ suc 𝑑 ) ) |
49 |
|
fin34i |
⊢ ( ( 𝐵 ∈ FinIII ∧ ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) : ω ⟶ 𝒫 𝐵 ∧ ∀ 𝑑 ∈ ω ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ 𝑑 ) ⊆ ( ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ‘ suc 𝑑 ) ) → ∪ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ∈ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ) |
50 |
1 13 48 49
|
syl3anc |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ∪ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ∈ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ) |
51 |
|
fin1a2lem11 |
⊢ ( ( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin ) → ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) ) |
52 |
51
|
adantrr |
⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) → ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) ) |
53 |
52
|
3ad2antl2 |
⊢ ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) → ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) ) |
54 |
53
|
adantr |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) ) |
55 |
|
simpll3 |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ¬ ∪ 𝐴 ∈ 𝐴 ) |
56 |
|
simplrr |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → 𝐴 ≠ ∅ ) |
57 |
|
sspwuni |
⊢ ( 𝐴 ⊆ 𝒫 ∅ ↔ ∪ 𝐴 ⊆ ∅ ) |
58 |
|
ss0b |
⊢ ( ∪ 𝐴 ⊆ ∅ ↔ ∪ 𝐴 = ∅ ) |
59 |
57 58
|
bitri |
⊢ ( 𝐴 ⊆ 𝒫 ∅ ↔ ∪ 𝐴 = ∅ ) |
60 |
|
pw0 |
⊢ 𝒫 ∅ = { ∅ } |
61 |
60
|
sseq2i |
⊢ ( 𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ { ∅ } ) |
62 |
|
sssn |
⊢ ( 𝐴 ⊆ { ∅ } ↔ ( 𝐴 = ∅ ∨ 𝐴 = { ∅ } ) ) |
63 |
61 62
|
bitri |
⊢ ( 𝐴 ⊆ 𝒫 ∅ ↔ ( 𝐴 = ∅ ∨ 𝐴 = { ∅ } ) ) |
64 |
|
df-ne |
⊢ ( 𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅ ) |
65 |
|
0ex |
⊢ ∅ ∈ V |
66 |
65
|
unisn |
⊢ ∪ { ∅ } = ∅ |
67 |
65
|
snid |
⊢ ∅ ∈ { ∅ } |
68 |
66 67
|
eqeltri |
⊢ ∪ { ∅ } ∈ { ∅ } |
69 |
|
unieq |
⊢ ( 𝐴 = { ∅ } → ∪ 𝐴 = ∪ { ∅ } ) |
70 |
|
id |
⊢ ( 𝐴 = { ∅ } → 𝐴 = { ∅ } ) |
71 |
69 70
|
eleq12d |
⊢ ( 𝐴 = { ∅ } → ( ∪ 𝐴 ∈ 𝐴 ↔ ∪ { ∅ } ∈ { ∅ } ) ) |
72 |
68 71
|
mpbiri |
⊢ ( 𝐴 = { ∅ } → ∪ 𝐴 ∈ 𝐴 ) |
73 |
72
|
orim2i |
⊢ ( ( 𝐴 = ∅ ∨ 𝐴 = { ∅ } ) → ( 𝐴 = ∅ ∨ ∪ 𝐴 ∈ 𝐴 ) ) |
74 |
73
|
ord |
⊢ ( ( 𝐴 = ∅ ∨ 𝐴 = { ∅ } ) → ( ¬ 𝐴 = ∅ → ∪ 𝐴 ∈ 𝐴 ) ) |
75 |
64 74
|
syl5bi |
⊢ ( ( 𝐴 = ∅ ∨ 𝐴 = { ∅ } ) → ( 𝐴 ≠ ∅ → ∪ 𝐴 ∈ 𝐴 ) ) |
76 |
63 75
|
sylbi |
⊢ ( 𝐴 ⊆ 𝒫 ∅ → ( 𝐴 ≠ ∅ → ∪ 𝐴 ∈ 𝐴 ) ) |
77 |
59 76
|
sylbir |
⊢ ( ∪ 𝐴 = ∅ → ( 𝐴 ≠ ∅ → ∪ 𝐴 ∈ 𝐴 ) ) |
78 |
77
|
com12 |
⊢ ( 𝐴 ≠ ∅ → ( ∪ 𝐴 = ∅ → ∪ 𝐴 ∈ 𝐴 ) ) |
79 |
78
|
con3d |
⊢ ( 𝐴 ≠ ∅ → ( ¬ ∪ 𝐴 ∈ 𝐴 → ¬ ∪ 𝐴 = ∅ ) ) |
80 |
56 55 79
|
sylc |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ¬ ∪ 𝐴 = ∅ ) |
81 |
|
ioran |
⊢ ( ¬ ( ∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅ ) ↔ ( ¬ ∪ 𝐴 ∈ 𝐴 ∧ ¬ ∪ 𝐴 = ∅ ) ) |
82 |
55 80 81
|
sylanbrc |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ¬ ( ∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅ ) ) |
83 |
|
uniun |
⊢ ∪ ( 𝐴 ∪ { ∅ } ) = ( ∪ 𝐴 ∪ ∪ { ∅ } ) |
84 |
66
|
uneq2i |
⊢ ( ∪ 𝐴 ∪ ∪ { ∅ } ) = ( ∪ 𝐴 ∪ ∅ ) |
85 |
|
un0 |
⊢ ( ∪ 𝐴 ∪ ∅ ) = ∪ 𝐴 |
86 |
83 84 85
|
3eqtri |
⊢ ∪ ( 𝐴 ∪ { ∅ } ) = ∪ 𝐴 |
87 |
86
|
eleq1i |
⊢ ( ∪ ( 𝐴 ∪ { ∅ } ) ∈ ( 𝐴 ∪ { ∅ } ) ↔ ∪ 𝐴 ∈ ( 𝐴 ∪ { ∅ } ) ) |
88 |
|
elun |
⊢ ( ∪ 𝐴 ∈ ( 𝐴 ∪ { ∅ } ) ↔ ( ∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 ∈ { ∅ } ) ) |
89 |
65
|
elsn2 |
⊢ ( ∪ 𝐴 ∈ { ∅ } ↔ ∪ 𝐴 = ∅ ) |
90 |
89
|
orbi2i |
⊢ ( ( ∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 ∈ { ∅ } ) ↔ ( ∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅ ) ) |
91 |
87 88 90
|
3bitri |
⊢ ( ∪ ( 𝐴 ∪ { ∅ } ) ∈ ( 𝐴 ∪ { ∅ } ) ↔ ( ∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅ ) ) |
92 |
82 91
|
sylnibr |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ¬ ∪ ( 𝐴 ∪ { ∅ } ) ∈ ( 𝐴 ∪ { ∅ } ) ) |
93 |
|
unieq |
⊢ ( ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) → ∪ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ∪ ( 𝐴 ∪ { ∅ } ) ) |
94 |
|
id |
⊢ ( ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) → ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) ) |
95 |
93 94
|
eleq12d |
⊢ ( ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) → ( ∪ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ∈ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ↔ ∪ ( 𝐴 ∪ { ∅ } ) ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
96 |
95
|
notbid |
⊢ ( ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) → ( ¬ ∪ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ∈ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ↔ ¬ ∪ ( 𝐴 ∪ { ∅ } ) ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
97 |
92 96
|
syl5ibrcom |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ( ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) = ( 𝐴 ∪ { ∅ } ) → ¬ ∪ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ∈ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ) ) |
98 |
54 97
|
mpd |
⊢ ( ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) ∧ 𝐵 ∈ FinIII ) → ¬ ∪ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ∈ ran ( 𝑒 ∈ ω ↦ ∪ { 𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒 } ) ) |
99 |
50 98
|
pm2.65da |
⊢ ( ( ( 𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴 ) ∧ ( 𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅ ) ) → ¬ 𝐵 ∈ FinIII ) |