nodenselem8

Description: Lemma for nodense . Give a condition for surreal less-than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011)

		Ref	Expression
	Assertion	nodenselem8	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( 𝐴 <s 𝐵 ↔ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) )

Proof

Step	Hyp	Ref	Expression
1		nodenselem5	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) )
2	1	exp32	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) → ( 𝐴 <s 𝐵 → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) ) )
3	2	3impia	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( 𝐴 <s 𝐵 → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) )
4		sltval2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1_o , ∅ ⟩ , ⟨ 1_o , 2_o ⟩ , ⟨ ∅ , 2_o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
5	4	3adant3	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1_o , ∅ ⟩ , ⟨ 1_o , 2_o ⟩ , ⟨ ∅ , 2_o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
6		fvex	⊢ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ V
7		fvex	⊢ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ V
8	6 7	brtp	⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1_o , ∅ ⟩ , ⟨ 1_o , 2_o ⟩ , ⟨ ∅ , 2_o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ↔ ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) )
9		eleq2	⊢ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ↔ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐵 ) ) )
10	9	biimpd	⊢ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐵 ) ) )
11		nosgnn0	⊢ ¬ ∅ ∈ { 1_o , 2_o }
12		nofnbday	⊢ ( 𝐵 ∈ No → 𝐵 Fn ( bday ‘ 𝐵 ) )
13		fnfvelrn	⊢ ( ( 𝐵 Fn ( bday ‘ 𝐵 ) ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐵 ) ) → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ ran 𝐵 )
14		eleq1	⊢ ( ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ → ( ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ ran 𝐵 ↔ ∅ ∈ ran 𝐵 ) )
15	13 14	syl5ibcom	⊢ ( ( 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	⊢ ( ( 𝐵 ∈ No ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐵 ) ) → ( ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ → ∅ ∈ ran 𝐵 ) )
17		norn	⊢ ( 𝐵 ∈ No → ran 𝐵 ⊆ { 1_o , 2_o } )
18	17	sseld	⊢ ( 𝐵 ∈ No → ( ∅ ∈ ran 𝐵 → ∅ ∈ { 1_o , 2_o } ) )
19	18	adantr	⊢ ( ( 𝐵 ∈ No ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐵 ) ) → ( ∅ ∈ ran 𝐵 → ∅ ∈ { 1_o , 2_o } ) )
20	16 19	syld	⊢ ( ( 𝐵 ∈ No ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐵 ) ) → ( ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ → ∅ ∈ { 1_o , 2_o } ) )
21	11 20	mtoi	⊢ ( ( 𝐵 ∈ No ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐵 ) ) → ¬ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ )
22	21	ex	⊢ ( 𝐵 ∈ No → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐵 ) → ¬ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) )
23	22	adantl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐵 ) → ¬ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) )
24	10 23	syl9r	⊢ ( ( 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	⊢ ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) → ¬ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) )
28		nofnbday	⊢ ( 𝐴 ∈ No → 𝐴 Fn ( bday ‘ 𝐴 ) )
29		fnfvelrn	⊢ ( ( 𝐴 Fn ( bday ‘ 𝐴 ) ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ ran 𝐴 )
30		eleq1	⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ ran 𝐴 ↔ ∅ ∈ ran 𝐴 ) )
31	29 30	syl5ibcom	⊢ ( ( 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	⊢ ( ( 𝐴 ∈ No ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ → ∅ ∈ ran 𝐴 ) )
33		norn	⊢ ( 𝐴 ∈ No → ran 𝐴 ⊆ { 1_o , 2_o } )
34	33	sseld	⊢ ( 𝐴 ∈ No → ( ∅ ∈ ran 𝐴 → ∅ ∈ { 1_o , 2_o } ) )
35	34	adantr	⊢ ( ( 𝐴 ∈ No ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) → ( ∅ ∈ ran 𝐴 → ∅ ∈ { 1_o , 2_o } ) )
36	32 35	syld	⊢ ( ( 𝐴 ∈ No ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ → ∅ ∈ { 1_o , 2_o } ) )
37	11 36	mtoi	⊢ ( ( 𝐴 ∈ No ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) → ¬ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ )
38	37	3ad2antl1	⊢ ( ( ( 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	⊢ ( ( ( 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 ) } ) = 2_o ) )
40		3orel13	⊢ ( ( ¬ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∧ ¬ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) → ( ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) )
41	27 39 40	syl2anc	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) → ( ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) )
42	41	ex	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) → ( ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) ) )
43	42	com23	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) ) )
44	8 43	biimtrid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1_o , ∅ ⟩ , ⟨ 1_o , 2_o ⟩ , ⟨ ∅ , 2_o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) ) )
45	5 44	sylbid	⊢ ( ( 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 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) ) )
46	3 45	mpdd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( 𝐴 <s 𝐵 → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) )
47		3mix2	⊢ ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) → ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) )
48	47 8	sylibr	⊢ ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1_o , ∅ ⟩ , ⟨ 1_o , 2_o ⟩ , ⟨ ∅ , 2_o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
49	48 5	imbitrrid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) → 𝐴 <s 𝐵 ) )
50	46 49	impbid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( 𝐴 <s 𝐵 ↔ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1_o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2_o ) ) )

Metamath Proof Explorer

Theorem nodenselem8

Proof