Step |
Hyp |
Ref |
Expression |
1 |
|
epweon |
⊢ E We On |
2 |
|
weso |
⊢ ( E We On → E Or On ) |
3 |
1 2
|
ax-mp |
⊢ E Or On |
4 |
|
soss |
⊢ ( 𝐴 ⊆ On → ( E Or On → E Or 𝐴 ) ) |
5 |
3 4
|
mpi |
⊢ ( 𝐴 ⊆ On → E Or 𝐴 ) |
6 |
|
fimax2g |
⊢ ( ( E Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ) |
7 |
5 6
|
syl3an1 |
⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ) |
8 |
|
ssel2 |
⊢ ( ( 𝐴 ⊆ On ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ On ) |
9 |
8
|
adantlr |
⊢ ( ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ On ) |
10 |
|
ssel2 |
⊢ ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ On ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ On ) |
12 |
|
epel |
⊢ ( 𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦 ) |
13 |
12
|
notbii |
⊢ ( ¬ 𝑥 E 𝑦 ↔ ¬ 𝑥 ∈ 𝑦 ) |
14 |
|
ontri1 |
⊢ ( ( 𝑦 ∈ On ∧ 𝑥 ∈ On ) → ( 𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦 ) ) |
15 |
13 14
|
bitr4id |
⊢ ( ( 𝑦 ∈ On ∧ 𝑥 ∈ On ) → ( ¬ 𝑥 E 𝑦 ↔ 𝑦 ⊆ 𝑥 ) ) |
16 |
9 11 15
|
syl2anc |
⊢ ( ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( ¬ 𝑥 E 𝑦 ↔ 𝑦 ⊆ 𝑥 ) ) |
17 |
16
|
ralbidva |
⊢ ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ) ) |
18 |
|
unissb |
⊢ ( ∪ 𝐴 ⊆ 𝑥 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ) |
19 |
17 18
|
bitr4di |
⊢ ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∪ 𝐴 ⊆ 𝑥 ) ) |
20 |
19
|
rexbidva |
⊢ ( 𝐴 ⊆ On → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∃ 𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥 ) ) |
21 |
20
|
3ad2ant1 |
⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 E 𝑦 ↔ ∃ 𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥 ) ) |
22 |
7 21
|
mpbid |
⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥 ) |
23 |
|
elssuni |
⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴 ) |
24 |
|
eqss |
⊢ ( 𝑥 = ∪ 𝐴 ↔ ( 𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥 ) ) |
25 |
|
eleq1 |
⊢ ( 𝑥 = ∪ 𝐴 → ( 𝑥 ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴 ) ) |
26 |
25
|
biimpcd |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝑥 = ∪ 𝐴 → ∪ 𝐴 ∈ 𝐴 ) ) |
27 |
24 26
|
syl5bir |
⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥 ) → ∪ 𝐴 ∈ 𝐴 ) ) |
28 |
23 27
|
mpand |
⊢ ( 𝑥 ∈ 𝐴 → ( ∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴 ) ) |
29 |
28
|
rexlimiv |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴 ) |
30 |
22 29
|
syl |
⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∪ 𝐴 ∈ 𝐴 ) |