Theorem nodenselem8 31841

Description: Lemma for nodense 31842. Give a condition for surreal less than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)

Assertion

Ref	Expression
nodenselem8	⊢ ((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)))

Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem8

Step	Hyp	Ref	Expression
1		nodenselem5 31838	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴))
2	1	exp32 631	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( bday ‘𝐴) = ( bday ‘𝐵) → (𝐴 <s 𝐵 → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴))))
3	2	3impia 1261	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)))
4		sltval2 31809	. . . . 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 1081	. . . 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 6201	. . . . . 6 ⊢ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ V
7		fvex 6201	. . . . . 6 ⊢ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ V
8	6, 7	brtp 31639	. . . . 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 2690	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝐴) = ( bday ‘𝐵) → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴) ↔ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)))
10	9	biimpd 219	. . . . . . . . . . . 12 ⊢ (( bday ‘𝐴) = ( bday ‘𝐵) → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)))
11		nosgnn0 31811	. . . . . . . . . . . . . . 15 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜}
12		nofnbday 31805	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ No → 𝐵 Fn ( bday ‘𝐵))
13		fnfvelrn 6356	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 Fn ( bday ‘𝐵) ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)) → (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐵)
14		eleq1 2689	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ((𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐵 ↔ ∅ ∈ ran 𝐵))
15	13, 14	syl5ibcom 235	. . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ No ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)) → ((𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈ ran 𝐵))
17		norn 31804	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ No → ran 𝐵 ⊆ {1𝑜, 2𝑜})
18	17	sseld 3602	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ No → (∅ ∈ ran 𝐵 → ∅ ∈ {1𝑜, 2𝑜}))
19	18	adantr 481	. . . . . . . . . . . . . . . 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 190	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ No ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)) → ¬ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅)
22	21	ex 450	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ No → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵) → ¬ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅))
23	22	adantl 482	. . . . . . . . . . . 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 1261	. . . . . . . . . 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 445	. . . . . . . . 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 962	. . . . . . . 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 31805	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴))
29		fnfvelrn 6356	. . . . . . . . . . . . . 14 ⊢ ((𝐴 Fn ( bday ‘𝐴) ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)) → (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐴)
30		eleq1 2689	. . . . . . . . . . . . . 14 ⊢ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐴 ↔ ∅ ∈ ran 𝐴))
31	29, 30	syl5ibcom 235	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)) → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈ ran 𝐴))
33		norn 31804	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
34	33	sseld 3602	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → (∅ ∈ ran 𝐴 → ∅ ∈ {1𝑜, 2𝑜}))
35	34	adantr 481	. . . . . . . . . . . 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 190	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)) → ¬ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅)
38	37	3ad2antl1 1223	. . . . . . . . 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 963	. . . . . . . 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 31598	. . . . . . . 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 693	. . . . . . 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 450	. . . . . 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	syl5bi 232	. . . 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 230	. . 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 1231	. . . 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 224	. . 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	syl5ibr 236	. 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 202	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)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {crab 2916  ∅c0 3915  {cpr 4179  {ctp 4181  ⟨cop 4183  ∩ cint 4475   class class class wbr 4653  ran crn 5115  Oncon0 5723   Fn wfn 5883  ‘cfv 5888  1_𝑜c1o 7553  2_𝑜c2o 7554   No csur 31793   <s cslt 31794   bday cbday 31795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798

This theorem is referenced by:  nodense  31842