| Step | Hyp | Ref
| Expression |
| 1 | | nodenselem6 27734 |
. 2
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ No
) |
| 2 | | bdayval 27693 |
. . . . 5
⊢ ((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ No
→ ( bday ‘(𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) = dom (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 3 | 1, 2 | syl 17 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ( bday
‘(𝐴 ↾
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) = dom (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 4 | | dmres 6030 |
. . . . 5
⊢ dom
(𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∩ dom 𝐴) |
| 5 | | nodenselem5 27733 |
. . . . . . . 8
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) |
| 6 | | bdayfo 27722 |
. . . . . . . . . . 11
⊢ bday : No –onto→On |
| 7 | | fof 6820 |
. . . . . . . . . . 11
⊢ ( bday : No –onto→On → bday
: No ⟶On) |
| 8 | 6, 7 | ax-mp 5 |
. . . . . . . . . 10
⊢ bday : No
⟶On |
| 9 | | 0elon 6438 |
. . . . . . . . . 10
⊢ ∅
∈ On |
| 10 | 8, 9 | f0cli 7118 |
. . . . . . . . 9
⊢ ( bday ‘𝐴) ∈ On |
| 11 | 10 | onelssi 6499 |
. . . . . . . 8
⊢ (∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴) →
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ ( bday
‘𝐴)) |
| 12 | 5, 11 | syl 17 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ ( bday
‘𝐴)) |
| 13 | | bdayval 27693 |
. . . . . . . 8
⊢ (𝐴 ∈
No → ( bday ‘𝐴) = dom 𝐴) |
| 14 | 13 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ( bday
‘𝐴) = dom
𝐴) |
| 15 | 12, 14 | sseqtrd 4020 |
. . . . . 6
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ dom 𝐴) |
| 16 | | dfss2 3969 |
. . . . . 6
⊢ (∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ dom 𝐴 ↔ (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∩ dom 𝐴) = ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) |
| 17 | 15, 16 | sylib 218 |
. . . . 5
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∩ dom 𝐴) = ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) |
| 18 | 4, 17 | eqtrid 2789 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → dom (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) |
| 19 | 3, 18 | eqtrd 2777 |
. . 3
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ( bday
‘(𝐴 ↾
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) = ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) |
| 20 | 19, 5 | eqeltrd 2841 |
. 2
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ( bday
‘(𝐴 ↾
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) ∈ ( bday
‘𝐴)) |
| 21 | | nodenselem4 27732 |
. . . . 5
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ 𝐴 <s 𝐵) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On) |
| 22 | 21 | adantrl 716 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On) |
| 23 | | nodenselem8 27736 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 ↔ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o))) |
| 24 | 23 | biimpd 229 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o))) |
| 25 | 24 | 3expia 1122 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → (( bday
‘𝐴) = ( bday ‘𝐵) → (𝐴 <s 𝐵 → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)))) |
| 26 | 25 | imp32 418 |
. . . . . . . . . 10
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)) |
| 27 | 26 | simpld 494 |
. . . . . . . . 9
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o) |
| 28 | | eqid 2737 |
. . . . . . . . 9
⊢ ∅ =
∅ |
| 29 | 27, 28 | jctir 520 |
. . . . . . . 8
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ ∅ =
∅)) |
| 30 | 29 | 3mix1d 1337 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ ∅ = ∅)
∨ ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ ∅ =
2o) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ ∅ =
2o))) |
| 31 | | fvex 6919 |
. . . . . . . 8
⊢ (𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ V |
| 32 | | 0ex 5307 |
. . . . . . . 8
⊢ ∅
∈ V |
| 33 | 31, 32 | brtp 5528 |
. . . . . . 7
⊢ ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅,
2o〉}∅ ↔ (((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ ∅ = ∅)
∨ ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ ∅ =
2o) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ ∅ =
2o))) |
| 34 | 30, 33 | sylibr 234 |
. . . . . 6
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅,
2o〉}∅) |
| 35 | 19 | fveq2d 6910 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘( bday
‘(𝐴 ↾
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 36 | | fvnobday 27723 |
. . . . . . . 8
⊢ ((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ No
→ ((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘( bday
‘(𝐴 ↾
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) = ∅) |
| 37 | 1, 36 | syl 17 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘( bday
‘(𝐴 ↾
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) = ∅) |
| 38 | 35, 37 | eqtr3d 2779 |
. . . . . 6
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) |
| 39 | 34, 38 | breqtrrd 5171 |
. . . . 5
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 40 | | fvres 6925 |
. . . . . . 7
⊢ (𝑦 ∈ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐴‘𝑦)) |
| 41 | 40 | eqcomd 2743 |
. . . . . 6
⊢ (𝑦 ∈ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦)) |
| 42 | 41 | rgen 3063 |
. . . . 5
⊢
∀𝑦 ∈
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) |
| 43 | 39, 42 | jctil 519 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) ∧ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) |
| 44 | | raleq 3323 |
. . . . . 6
⊢ (𝑥 = ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) ↔ ∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦))) |
| 45 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → (𝐴‘𝑥) = (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 46 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 47 | 45, 46 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → ((𝐴‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥) ↔ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) |
| 48 | 44, 47 | anbi12d 632 |
. . . . 5
⊢ (𝑥 = ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → ((∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) ∧ (𝐴‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥)) ↔ (∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) ∧ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})))) |
| 49 | 48 | rspcev 3622 |
. . . 4
⊢ ((∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On ∧ (∀𝑦 ∈ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) ∧ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) ∧ (𝐴‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥))) |
| 50 | 22, 43, 49 | syl2anc 584 |
. . 3
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) ∧ (𝐴‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥))) |
| 51 | | simpll 767 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 ∈ No
) |
| 52 | | sltval 27692 |
. . . 4
⊢ ((𝐴 ∈
No ∧ (𝐴 ↾
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ No )
→ (𝐴 <s (𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) ∧ (𝐴‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥)))) |
| 53 | 51, 1, 52 | syl2anc 584 |
. . 3
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 <s (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) ∧ (𝐴‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥)))) |
| 54 | 50, 53 | mpbird 257 |
. 2
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 <s (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 55 | 41 | adantl 481 |
. . . . . 6
⊢ ((((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) → (𝐴‘𝑦) = ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦)) |
| 56 | | nodenselem7 27735 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → (𝐴‘𝑦) = (𝐵‘𝑦))) |
| 57 | 56 | imp 406 |
. . . . . 6
⊢ ((((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) → (𝐴‘𝑦) = (𝐵‘𝑦)) |
| 58 | 55, 57 | eqtr3d 2779 |
. . . . 5
⊢ ((((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) → ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦)) |
| 59 | 58 | ralrimiva 3146 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦)) |
| 60 | 26 | simprd 495 |
. . . . . . . 8
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) |
| 61 | 60, 28 | jctil 519 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (∅ = ∅ ∧ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)) |
| 62 | 61 | 3mix3d 1339 |
. . . . . 6
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ((∅ = 1o ∧
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) ∨ (∅ = 1o
∧ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) ∨ (∅ = ∅
∧ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o))) |
| 63 | | fvex 6919 |
. . . . . . 7
⊢ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ V |
| 64 | 32, 63 | brtp 5528 |
. . . . . 6
⊢
(∅{〈1o, ∅〉, 〈1o,
2o〉, 〈∅, 2o〉} (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ↔ ((∅ = 1o ∧
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) ∨ (∅ = 1o
∧ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) ∨ (∅ = ∅
∧ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o))) |
| 65 | 62, 64 | sylibr 234 |
. . . . 5
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∅{〈1o,
∅〉, 〈1o, 2o〉, 〈∅,
2o〉} (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 66 | 38, 65 | eqbrtrd 5165 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 67 | | raleq 3323 |
. . . . . 6
⊢ (𝑥 = ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → (∀𝑦 ∈ 𝑥 ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦) ↔ ∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦))) |
| 68 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → (𝐵‘𝑥) = (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) |
| 69 | 46, 68 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → (((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘𝑥) ↔ ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) |
| 70 | 67, 69 | anbi12d 632 |
. . . . 5
⊢ (𝑥 = ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → ((∀𝑦 ∈ 𝑥 ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦) ∧ ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘𝑥)) ↔ (∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦) ∧ ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})))) |
| 71 | 70 | rspcev 3622 |
. . . 4
⊢ ((∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On ∧ (∀𝑦 ∈ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦) ∧ ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦) ∧ ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘𝑥))) |
| 72 | 22, 59, 66, 71 | syl12anc 837 |
. . 3
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦) ∧ ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘𝑥))) |
| 73 | | simplr 769 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → 𝐵 ∈ No
) |
| 74 | | sltval 27692 |
. . . 4
⊢ (((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ No
∧ 𝐵 ∈ No ) → ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦) ∧ ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘𝑥)))) |
| 75 | 1, 73, 74 | syl2anc 584 |
. . 3
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑦) = (𝐵‘𝑦) ∧ ((𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})‘𝑥){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘𝑥)))) |
| 76 | 72, 75 | mpbird 257 |
. 2
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) <s 𝐵) |
| 77 | | fveq2 6906 |
. . . . 5
⊢ (𝑥 = (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) → ( bday
‘𝑥) = ( bday ‘(𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) |
| 78 | 77 | eleq1d 2826 |
. . . 4
⊢ (𝑥 = (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) → (( bday
‘𝑥) ∈
( bday ‘𝐴) ↔ ( bday
‘(𝐴 ↾
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) ∈ ( bday
‘𝐴))) |
| 79 | | breq2 5147 |
. . . 4
⊢ (𝑥 = (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) → (𝐴 <s 𝑥 ↔ 𝐴 <s (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) |
| 80 | | breq1 5146 |
. . . 4
⊢ (𝑥 = (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) → (𝑥 <s 𝐵 ↔ (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) <s 𝐵)) |
| 81 | 78, 79, 80 | 3anbi123d 1438 |
. . 3
⊢ (𝑥 = (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) → ((( bday
‘𝑥) ∈
( bday ‘𝐴) ∧ 𝐴 <s 𝑥 ∧ 𝑥 <s 𝐵) ↔ (( bday
‘(𝐴 ↾
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) ∈ ( bday
‘𝐴) ∧
𝐴 <s (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∧ (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) <s 𝐵))) |
| 82 | 81 | rspcev 3622 |
. 2
⊢ (((𝐴 ↾ ∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ No
∧ (( bday ‘(𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) ∈ ( bday
‘𝐴) ∧
𝐴 <s (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∧ (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) <s 𝐵)) → ∃𝑥 ∈ No
(( bday ‘𝑥) ∈ ( bday
‘𝐴) ∧
𝐴 <s 𝑥 ∧ 𝑥 <s 𝐵)) |
| 83 | 1, 20, 54, 76, 82 | syl13anc 1374 |
1
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 ∈ No
(( bday ‘𝑥) ∈ ( bday
‘𝐴) ∧
𝐴 <s 𝑥 ∧ 𝑥 <s 𝐵)) |