Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
⊢ ( 𝐴 ∈ P → 𝐴 ∈ V ) |
2 |
|
prnmax |
⊢ ( ( 𝐴 ∈ P ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ) |
3 |
2
|
ralrimiva |
⊢ ( 𝐴 ∈ P → ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ) |
4 |
|
prpssnq |
⊢ ( 𝐴 ∈ P → 𝐴 ⊊ Q ) |
5 |
4
|
pssssd |
⊢ ( 𝐴 ∈ P → 𝐴 ⊆ Q ) |
6 |
|
ltsonq |
⊢ <Q Or Q |
7 |
|
soss |
⊢ ( 𝐴 ⊆ Q → ( <Q Or Q → <Q Or 𝐴 ) ) |
8 |
5 6 7
|
mpisyl |
⊢ ( 𝐴 ∈ P → <Q Or 𝐴 ) |
9 |
8
|
adantr |
⊢ ( ( 𝐴 ∈ P ∧ 𝐴 ∈ Fin ) → <Q Or 𝐴 ) |
10 |
|
simpr |
⊢ ( ( 𝐴 ∈ P ∧ 𝐴 ∈ Fin ) → 𝐴 ∈ Fin ) |
11 |
|
prn0 |
⊢ ( 𝐴 ∈ P → 𝐴 ≠ ∅ ) |
12 |
11
|
adantr |
⊢ ( ( 𝐴 ∈ P ∧ 𝐴 ∈ Fin ) → 𝐴 ≠ ∅ ) |
13 |
|
fimax2g |
⊢ ( ( <Q Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ) |
14 |
9 10 12 13
|
syl3anc |
⊢ ( ( 𝐴 ∈ P ∧ 𝐴 ∈ Fin ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ) |
15 |
|
ralnex |
⊢ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃ 𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ) |
16 |
15
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃ 𝑥 ∈ 𝐴 ¬ ∃ 𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ) |
17 |
|
rexnal |
⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ ∃ 𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ) |
18 |
16 17
|
bitri |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ) |
19 |
14 18
|
sylib |
⊢ ( ( 𝐴 ∈ P ∧ 𝐴 ∈ Fin ) → ¬ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ) |
20 |
19
|
ex |
⊢ ( 𝐴 ∈ P → ( 𝐴 ∈ Fin → ¬ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ) ) |
21 |
3 20
|
mt2d |
⊢ ( 𝐴 ∈ P → ¬ 𝐴 ∈ Fin ) |
22 |
|
nelne1 |
⊢ ( ( 𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin ) → V ≠ Fin ) |
23 |
1 21 22
|
syl2anc |
⊢ ( 𝐴 ∈ P → V ≠ Fin ) |
24 |
23
|
necomd |
⊢ ( 𝐴 ∈ P → Fin ≠ V ) |
25 |
|
fineqv |
⊢ ( ¬ ω ∈ V ↔ Fin = V ) |
26 |
25
|
necon1abii |
⊢ ( Fin ≠ V ↔ ω ∈ V ) |
27 |
24 26
|
sylib |
⊢ ( 𝐴 ∈ P → ω ∈ V ) |