Proof of Theorem nodenselem8
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nodenselem5 27734 | . . . . 5
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (( bday
‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) | 
| 2 | 1 | exp32 420 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (( bday
‘𝐴) = ( bday ‘𝐵) → (𝐴 <s 𝐵 → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)))) | 
| 3 | 2 | 3impia 1117 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴))) | 
| 4 |  | sltval2 27702 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (𝐴 <s 𝐵 ↔ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) | 
| 5 | 4 | 3adant3 1132 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 ↔ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}))) | 
| 6 |  | fvex 6918 | . . . . . 6
⊢ (𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ V | 
| 7 |  | fvex 6918 | . . . . . 6
⊢ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ V | 
| 8 | 6, 7 | brtp 5527 | . . . . 5
⊢ ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ↔ (((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o))) | 
| 9 |  | eleq2 2829 | . . . . . . . . . . . . 13
⊢ (( bday ‘𝐴) = ( bday
‘𝐵) →
(∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴) ↔
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵))) | 
| 10 | 9 | biimpd 229 | . . . . . . . . . . . 12
⊢ (( bday ‘𝐴) = ( bday
‘𝐵) →
(∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴) →
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵))) | 
| 11 |  | nosgnn0 27704 | . . . . . . . . . . . . . . 15
⊢  ¬
∅ ∈ {1o, 2o} | 
| 12 |  | nofnbday 27698 | . . . . . . . . . . . . . . . . 17
⊢ (𝐵 ∈ 
No  → 𝐵 Fn
( bday ‘𝐵)) | 
| 13 |  | fnfvelrn 7099 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐵 Fn ( bday
‘𝐵) ∧
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵)) →
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐵) | 
| 14 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ((𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐵 ↔ ∅ ∈ ran 𝐵)) | 
| 15 | 13, 14 | syl5ibcom 245 | . . . . . . . . . . . . . . . . 17
⊢ ((𝐵 Fn ( bday
‘𝐵) ∧
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵)) →
((𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈ ran 𝐵)) | 
| 16 | 12, 15 | sylan 580 | . . . . . . . . . . . . . . . 16
⊢ ((𝐵 ∈ 
No  ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵)) →
((𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈ ran 𝐵)) | 
| 17 |  | norn 27697 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ 
No  → ran 𝐵
⊆ {1o, 2o}) | 
| 18 | 17 | sseld 3981 | . . . . . . . . . . . . . . . . 17
⊢ (𝐵 ∈ 
No  → (∅ ∈ ran 𝐵 → ∅ ∈ {1o,
2o})) | 
| 19 | 18 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝐵 ∈ 
No  ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵)) →
(∅ ∈ ran 𝐵
→ ∅ ∈ {1o, 2o})) | 
| 20 | 16, 19 | syld 47 | . . . . . . . . . . . . . . 15
⊢ ((𝐵 ∈ 
No  ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵)) →
((𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈
{1o, 2o})) | 
| 21 | 11, 20 | mtoi 199 | . . . . . . . . . . . . . 14
⊢ ((𝐵 ∈ 
No  ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵)) →
¬ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) | 
| 22 | 21 | ex 412 | . . . . . . . . . . . . 13
⊢ (𝐵 ∈ 
No  → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵) →
¬ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅)) | 
| 23 | 22 | adantl 481 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐵) →
¬ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅)) | 
| 24 | 10, 23 | syl9r 78 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (( bday
‘𝐴) = ( bday ‘𝐵) → (∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴) →
¬ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅))) | 
| 25 | 24 | 3impia 1117 | . . . . . . . . . 10
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → (∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴) →
¬ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅)) | 
| 26 | 25 | imp 406 | . . . . . . . . 9
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
¬ (𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) | 
| 27 | 26 | intnand 488 | . . . . . . . 8
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
¬ ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅)) | 
| 28 |  | nofnbday 27698 | . . . . . . . . . . . . 13
⊢ (𝐴 ∈ 
No  → 𝐴 Fn
( bday ‘𝐴)) | 
| 29 |  | fnfvelrn 7099 | . . . . . . . . . . . . . 14
⊢ ((𝐴 Fn ( bday
‘𝐴) ∧
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
(𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐴) | 
| 30 |  | eleq1 2828 | . . . . . . . . . . . . . 14
⊢ ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐴 ↔ ∅ ∈ ran 𝐴)) | 
| 31 | 29, 30 | syl5ibcom 245 | . . . . . . . . . . . . 13
⊢ ((𝐴 Fn ( bday
‘𝐴) ∧
∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈ ran 𝐴)) | 
| 32 | 28, 31 | sylan 580 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ 
No  ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈ ran 𝐴)) | 
| 33 |  | norn 27697 | . . . . . . . . . . . . . 14
⊢ (𝐴 ∈ 
No  → ran 𝐴
⊆ {1o, 2o}) | 
| 34 | 33 | sseld 3981 | . . . . . . . . . . . . 13
⊢ (𝐴 ∈ 
No  → (∅ ∈ ran 𝐴 → ∅ ∈ {1o,
2o})) | 
| 35 | 34 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ 
No  ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
(∅ ∈ ran 𝐴
→ ∅ ∈ {1o, 2o})) | 
| 36 | 32, 35 | syld 47 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ 
No  ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈
{1o, 2o})) | 
| 37 | 11, 36 | mtoi 199 | . . . . . . . . . 10
⊢ ((𝐴 ∈ 
No  ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
¬ (𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) | 
| 38 | 37 | 3ad2antl1 1185 | . . . . . . . . 9
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
¬ (𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) | 
| 39 | 38 | intnanrd 489 | . . . . . . . 8
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
¬ ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)) | 
| 40 |  | 3orel13 1488 | . . . . . . . 8
⊢ ((¬
((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) ∧ ¬ ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)) → ((((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)) → ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o))) | 
| 41 | 27, 39, 40 | syl2anc 584 | . . . . . . 7
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴)) →
((((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)) → ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o))) | 
| 42 | 41 | ex 412 | . . . . . 6
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → (∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴) →
((((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)) → ((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)))) | 
| 43 | 42 | com23 86 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → ((((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)) → (∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴) →
((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)))) | 
| 44 | 8, 43 | biimtrid 242 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) → (∩
{𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴) →
((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)))) | 
| 45 | 5, 44 | sylbid 240 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday
‘𝐴) →
((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)))) | 
| 46 | 3, 45 | mpdd 43 | . 2
⊢ ((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))) | 
| 47 |  | 3mix2 1331 | . . . 4
⊢ (((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) → (((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o))) | 
| 48 | 47, 8 | sylibr 234 | . . 3
⊢ (((𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) → (𝐴‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑎
∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})) | 
| 49 | 48, 5 | imbitrrid 246 | . 2
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ ( bday
‘𝐴) = ( bday ‘𝐵)) → (((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o) → 𝐴 <s 𝐵)) | 
| 50 | 46, 49 | impbid 212 | 1
⊢ ((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))) |