Theorem nodenselem8 33308

Description: Lemma for nodense 33309. 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 33305	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴))
2	1	exp32 424	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( bday ‘𝐴) = ( bday ‘𝐵) → (𝐴 <s 𝐵 → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴))))
3	2	3impia 1114	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)))
4		sltval2 33276	. . . . 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 1129	. . . 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 6658	. . . . . 6 ⊢ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ V
7		fvex 6658	. . . . . 6 ⊢ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ V
8	6, 7	brtp 33098	. . . . 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 2878	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝐴) = ( bday ‘𝐵) → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴) ↔ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)))
10	9	biimpd 232	. . . . . . . . . . . 12 ⊢ (( bday ‘𝐴) = ( bday ‘𝐵) → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)))
11		nosgnn0 33278	. . . . . . . . . . . . . . 15 ⊢ ¬ ∅ ∈ {1o, 2o}
12		nofnbday 33272	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ No → 𝐵 Fn ( bday ‘𝐵))
13		fnfvelrn 6825	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 Fn ( bday ‘𝐵) ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)) → (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐵)
14		eleq1 2877	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ((𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐵 ↔ ∅ ∈ ran 𝐵))
15	13, 14	syl5ibcom 248	. . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ No ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)) → ((𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈ ran 𝐵))
17		norn 33271	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ No → ran 𝐵 ⊆ {1o, 2o})
18	17	sseld 3914	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ No → (∅ ∈ ran 𝐵 → ∅ ∈ {1o, 2o}))
19	18	adantr 484	. . . . . . . . . . . . . . . 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 202	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ No ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵)) → ¬ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅)
22	21	ex 416	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ No → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐵) → ¬ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅))
23	22	adantl 485	. . . . . . . . . . . 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 1114	. . . . . . . . . 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 410	. . . . . . . . 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 492	. . . . . . . 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 33272	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴))
29		fnfvelrn 6825	. . . . . . . . . . . . . 14 ⊢ ((𝐴 Fn ( bday ‘𝐴) ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)) → (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐴)
30		eleq1 2877	. . . . . . . . . . . . . 14 ⊢ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ ran 𝐴 ↔ ∅ ∈ ran 𝐴))
31	29, 30	syl5ibcom 248	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)) → ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅ → ∅ ∈ ran 𝐴))
33		norn 33271	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
34	33	sseld 3914	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → (∅ ∈ ran 𝐴 → ∅ ∈ {1o, 2o}))
35	34	adantr 484	. . . . . . . . . . . 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 202	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)) → ¬ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = ∅)
38	37	3ad2antl1 1182	. . . . . . . . 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 493	. . . . . . . 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 33060	. . . . . . . 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 587	. . . . . . 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 416	. . . . . 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 245	. . . 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 243	. . 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 1328	. . . 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 237	. . 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 249	. 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 215	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 209   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  ∅c0 4243  {cpr 4527  {ctp 4529  ⟨cop 4531  ∩ cint 4838   class class class wbr 5030  ran crn 5520  Oncon0 6159   Fn wfn 6319  ‘cfv 6324  1_oc1o 8078  2_oc2o 8079   No csur 33260   <s cslt 33261   bday cbday 33262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1o 8085  df-2o 8086  df-no 33263  df-slt 33264  df-bday 33265

This theorem is referenced by:  nodense  33309