Step |
Hyp |
Ref |
Expression |
1 |
|
simp2 |
⊢ ( ( 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) ) → 𝐴 ≠ ∅ ) |
2 |
|
simp1 |
⊢ ( ( 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) ) → 𝐴 ∈ dom card ) |
3 |
|
snfi |
⊢ { ∅ } ∈ Fin |
4 |
|
finnum |
⊢ ( { ∅ } ∈ Fin → { ∅ } ∈ dom card ) |
5 |
3 4
|
ax-mp |
⊢ { ∅ } ∈ dom card |
6 |
|
unnum |
⊢ ( ( 𝐴 ∈ dom card ∧ { ∅ } ∈ dom card ) → ( 𝐴 ∪ { ∅ } ) ∈ dom card ) |
7 |
2 5 6
|
sylancl |
⊢ ( ( 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) ) → ( 𝐴 ∪ { ∅ } ) ∈ dom card ) |
8 |
|
uncom |
⊢ ( 𝐴 ∪ { ∅ } ) = ( { ∅ } ∪ 𝐴 ) |
9 |
8
|
sseq2i |
⊢ ( 𝑤 ⊆ ( 𝐴 ∪ { ∅ } ) ↔ 𝑤 ⊆ ( { ∅ } ∪ 𝐴 ) ) |
10 |
|
ssundif |
⊢ ( 𝑤 ⊆ ( { ∅ } ∪ 𝐴 ) ↔ ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ) |
11 |
9 10
|
bitri |
⊢ ( 𝑤 ⊆ ( 𝐴 ∪ { ∅ } ) ↔ ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ) |
12 |
|
difss |
⊢ ( 𝑤 ∖ { ∅ } ) ⊆ 𝑤 |
13 |
|
soss |
⊢ ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝑤 → ( [⊊] Or 𝑤 → [⊊] Or ( 𝑤 ∖ { ∅ } ) ) ) |
14 |
12 13
|
ax-mp |
⊢ ( [⊊] Or 𝑤 → [⊊] Or ( 𝑤 ∖ { ∅ } ) ) |
15 |
|
ssdif0 |
⊢ ( 𝑤 ⊆ { ∅ } ↔ ( 𝑤 ∖ { ∅ } ) = ∅ ) |
16 |
|
uni0b |
⊢ ( ∪ 𝑤 = ∅ ↔ 𝑤 ⊆ { ∅ } ) |
17 |
16
|
biimpri |
⊢ ( 𝑤 ⊆ { ∅ } → ∪ 𝑤 = ∅ ) |
18 |
17
|
eleq1d |
⊢ ( 𝑤 ⊆ { ∅ } → ( ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ↔ ∅ ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
19 |
15 18
|
sylbir |
⊢ ( ( 𝑤 ∖ { ∅ } ) = ∅ → ( ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ↔ ∅ ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
20 |
19
|
imbi2d |
⊢ ( ( 𝑤 ∖ { ∅ } ) = ∅ → ( ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) ↔ ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∅ ∈ ( 𝐴 ∪ { ∅ } ) ) ) ) |
21 |
|
vex |
⊢ 𝑤 ∈ V |
22 |
21
|
difexi |
⊢ ( 𝑤 ∖ { ∅ } ) ∈ V |
23 |
|
sseq1 |
⊢ ( 𝑧 = ( 𝑤 ∖ { ∅ } ) → ( 𝑧 ⊆ 𝐴 ↔ ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ) ) |
24 |
|
neeq1 |
⊢ ( 𝑧 = ( 𝑤 ∖ { ∅ } ) → ( 𝑧 ≠ ∅ ↔ ( 𝑤 ∖ { ∅ } ) ≠ ∅ ) ) |
25 |
|
soeq2 |
⊢ ( 𝑧 = ( 𝑤 ∖ { ∅ } ) → ( [⊊] Or 𝑧 ↔ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) ) |
26 |
23 24 25
|
3anbi123d |
⊢ ( 𝑧 = ( 𝑤 ∖ { ∅ } ) → ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) ↔ ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ ( 𝑤 ∖ { ∅ } ) ≠ ∅ ∧ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) ) ) |
27 |
|
unieq |
⊢ ( 𝑧 = ( 𝑤 ∖ { ∅ } ) → ∪ 𝑧 = ∪ ( 𝑤 ∖ { ∅ } ) ) |
28 |
27
|
eleq1d |
⊢ ( 𝑧 = ( 𝑤 ∖ { ∅ } ) → ( ∪ 𝑧 ∈ 𝐴 ↔ ∪ ( 𝑤 ∖ { ∅ } ) ∈ 𝐴 ) ) |
29 |
26 28
|
imbi12d |
⊢ ( 𝑧 = ( 𝑤 ∖ { ∅ } ) → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) ↔ ( ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ ( 𝑤 ∖ { ∅ } ) ≠ ∅ ∧ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) → ∪ ( 𝑤 ∖ { ∅ } ) ∈ 𝐴 ) ) ) |
30 |
22 29
|
spcv |
⊢ ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ( ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ ( 𝑤 ∖ { ∅ } ) ≠ ∅ ∧ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) → ∪ ( 𝑤 ∖ { ∅ } ) ∈ 𝐴 ) ) |
31 |
30
|
com12 |
⊢ ( ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ ( 𝑤 ∖ { ∅ } ) ≠ ∅ ∧ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) → ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∪ ( 𝑤 ∖ { ∅ } ) ∈ 𝐴 ) ) |
32 |
31
|
3expa |
⊢ ( ( ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ ( 𝑤 ∖ { ∅ } ) ≠ ∅ ) ∧ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) → ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∪ ( 𝑤 ∖ { ∅ } ) ∈ 𝐴 ) ) |
33 |
32
|
an32s |
⊢ ( ( ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) ∧ ( 𝑤 ∖ { ∅ } ) ≠ ∅ ) → ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∪ ( 𝑤 ∖ { ∅ } ) ∈ 𝐴 ) ) |
34 |
|
unidif0 |
⊢ ∪ ( 𝑤 ∖ { ∅ } ) = ∪ 𝑤 |
35 |
34
|
eleq1i |
⊢ ( ∪ ( 𝑤 ∖ { ∅ } ) ∈ 𝐴 ↔ ∪ 𝑤 ∈ 𝐴 ) |
36 |
|
elun1 |
⊢ ( ∪ 𝑤 ∈ 𝐴 → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) |
37 |
35 36
|
sylbi |
⊢ ( ∪ ( 𝑤 ∖ { ∅ } ) ∈ 𝐴 → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) |
38 |
33 37
|
syl6 |
⊢ ( ( ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) ∧ ( 𝑤 ∖ { ∅ } ) ≠ ∅ ) → ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
39 |
|
0ex |
⊢ ∅ ∈ V |
40 |
39
|
snid |
⊢ ∅ ∈ { ∅ } |
41 |
|
elun2 |
⊢ ( ∅ ∈ { ∅ } → ∅ ∈ ( 𝐴 ∪ { ∅ } ) ) |
42 |
40 41
|
ax-mp |
⊢ ∅ ∈ ( 𝐴 ∪ { ∅ } ) |
43 |
42
|
2a1i |
⊢ ( ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) → ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∅ ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
44 |
20 38 43
|
pm2.61ne |
⊢ ( ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ [⊊] Or ( 𝑤 ∖ { ∅ } ) ) → ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
45 |
14 44
|
sylan2 |
⊢ ( ( ( 𝑤 ∖ { ∅ } ) ⊆ 𝐴 ∧ [⊊] Or 𝑤 ) → ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
46 |
11 45
|
sylanb |
⊢ ( ( 𝑤 ⊆ ( 𝐴 ∪ { ∅ } ) ∧ [⊊] Or 𝑤 ) → ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
47 |
46
|
com12 |
⊢ ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ( ( 𝑤 ⊆ ( 𝐴 ∪ { ∅ } ) ∧ [⊊] Or 𝑤 ) → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
48 |
47
|
alrimiv |
⊢ ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) → ∀ 𝑤 ( ( 𝑤 ⊆ ( 𝐴 ∪ { ∅ } ) ∧ [⊊] Or 𝑤 ) → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
49 |
48
|
3ad2ant3 |
⊢ ( ( 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) ) → ∀ 𝑤 ( ( 𝑤 ⊆ ( 𝐴 ∪ { ∅ } ) ∧ [⊊] Or 𝑤 ) → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) ) |
50 |
|
zorng |
⊢ ( ( ( 𝐴 ∪ { ∅ } ) ∈ dom card ∧ ∀ 𝑤 ( ( 𝑤 ⊆ ( 𝐴 ∪ { ∅ } ) ∧ [⊊] Or 𝑤 ) → ∪ 𝑤 ∈ ( 𝐴 ∪ { ∅ } ) ) ) → ∃ 𝑥 ∈ ( 𝐴 ∪ { ∅ } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { ∅ } ) ¬ 𝑥 ⊊ 𝑦 ) |
51 |
7 49 50
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) ) → ∃ 𝑥 ∈ ( 𝐴 ∪ { ∅ } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { ∅ } ) ¬ 𝑥 ⊊ 𝑦 ) |
52 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ { ∅ } ) |
53 |
|
ssralv |
⊢ ( 𝐴 ⊆ ( 𝐴 ∪ { ∅ } ) → ( ∀ 𝑦 ∈ ( 𝐴 ∪ { ∅ } ) ¬ 𝑥 ⊊ 𝑦 → ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) ) |
54 |
52 53
|
ax-mp |
⊢ ( ∀ 𝑦 ∈ ( 𝐴 ∪ { ∅ } ) ¬ 𝑥 ⊊ 𝑦 → ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) |
55 |
54
|
reximi |
⊢ ( ∃ 𝑥 ∈ ( 𝐴 ∪ { ∅ } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { ∅ } ) ¬ 𝑥 ⊊ 𝑦 → ∃ 𝑥 ∈ ( 𝐴 ∪ { ∅ } ) ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) |
56 |
|
rexun |
⊢ ( ∃ 𝑥 ∈ ( 𝐴 ∪ { ∅ } ) ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃ 𝑥 ∈ { ∅ } ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) ) |
57 |
|
simpr |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) |
58 |
|
simpr |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ { ∅ } ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) → ∃ 𝑥 ∈ { ∅ } ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) |
59 |
|
psseq1 |
⊢ ( 𝑥 = ∅ → ( 𝑥 ⊊ 𝑦 ↔ ∅ ⊊ 𝑦 ) ) |
60 |
|
0pss |
⊢ ( ∅ ⊊ 𝑦 ↔ 𝑦 ≠ ∅ ) |
61 |
59 60
|
bitrdi |
⊢ ( 𝑥 = ∅ → ( 𝑥 ⊊ 𝑦 ↔ 𝑦 ≠ ∅ ) ) |
62 |
61
|
notbid |
⊢ ( 𝑥 = ∅ → ( ¬ 𝑥 ⊊ 𝑦 ↔ ¬ 𝑦 ≠ ∅ ) ) |
63 |
|
nne |
⊢ ( ¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅ ) |
64 |
62 63
|
bitrdi |
⊢ ( 𝑥 = ∅ → ( ¬ 𝑥 ⊊ 𝑦 ↔ 𝑦 = ∅ ) ) |
65 |
64
|
ralbidv |
⊢ ( 𝑥 = ∅ → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 = ∅ ) ) |
66 |
39 65
|
rexsn |
⊢ ( ∃ 𝑥 ∈ { ∅ } ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 = ∅ ) |
67 |
|
eqsn |
⊢ ( 𝐴 ≠ ∅ → ( 𝐴 = { ∅ } ↔ ∀ 𝑦 ∈ 𝐴 𝑦 = ∅ ) ) |
68 |
67
|
biimpar |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑦 ∈ 𝐴 𝑦 = ∅ ) → 𝐴 = { ∅ } ) |
69 |
66 68
|
sylan2b |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ { ∅ } ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) → 𝐴 = { ∅ } ) |
70 |
69
|
rexeqdv |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ { ∅ } ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∃ 𝑥 ∈ { ∅ } ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) ) |
71 |
58 70
|
mpbird |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ { ∅ } ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) |
72 |
57 71
|
jaodan |
⊢ ( ( 𝐴 ≠ ∅ ∧ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃ 𝑥 ∈ { ∅ } ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) |
73 |
56 72
|
sylan2b |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ( 𝐴 ∪ { ∅ } ) ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) |
74 |
55 73
|
sylan2 |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ( 𝐴 ∪ { ∅ } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { ∅ } ) ¬ 𝑥 ⊊ 𝑦 ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) |
75 |
1 51 74
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧 ) → ∪ 𝑧 ∈ 𝐴 ) ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ) |