Theorem madebday 33663

Description: A surreal is part of the set made by ordinal 𝐴 iff its birthday is less than or equal to 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)

Assertion

Ref	Expression
madebday	⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))

Proof of Theorem madebday

Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		madebdayim 33653	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ ( M ‘𝐴)) → ( bday ‘𝑋) ⊆ 𝐴)
2	1	ex 416	. . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴))
3	2	adantr 484	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴))
4		sseq2 3920	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑏))
5		fveq2 6662	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
6	5	eleq2d 2837	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝑏)))
7	4, 6	imbi12d 348	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏))))
8	7	ralbidv 3126	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏))))
9		fveq2 6662	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
10	9	sseq1d 3925	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑥) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ 𝑏))
11		eleq1 2839	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘𝑏)))
12	10, 11	imbi12d 348	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏)) ↔ (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))))
13	12	cbvralvw 3361	. . . . 5 ⊢ (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏)) ↔ ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
14	8, 13	bitrdi 290	. . . 4 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))))
15		sseq2 3920	. . . . . 6 ⊢ (𝑎 = 𝐴 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝐴))
16		fveq2 6662	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
17	16	eleq2d 2837	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝐴)))
18	15, 17	imbi12d 348	. . . . 5 ⊢ (𝑎 = 𝐴 → ((( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴))))
19	18	ralbidv 3126	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴))))
20		bdayelon 33560	. . . . . . . . 9 ⊢ ( bday ‘𝑥) ∈ On
21		onsseleq 6214	. . . . . . . . 9 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝑎 ∈ On) → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
22	20, 21	mpan 689	. . . . . . . 8 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
23	22	ad2antrr 725	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
24		simpll 766	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → 𝑎 ∈ On)
25		onelss 6215	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ∈ 𝑎 → ( bday ‘𝑥) ⊆ 𝑎))
26	25	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ∈ 𝑎 → ( bday ‘𝑥) ⊆ 𝑎))
27	26	imp 410	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( bday ‘𝑥) ⊆ 𝑎)
28		madess 33642	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝑎 ∈ On ∧ ( bday ‘𝑥) ⊆ 𝑎) → ( M ‘( bday ‘𝑥)) ⊆ ( M ‘𝑎))
29	20, 24, 27, 28	mp3an2ani 1465	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( M ‘( bday ‘𝑥)) ⊆ ( M ‘𝑎))
30		ssid 3916	. . . . . . . . . . 11 ⊢ ( bday ‘𝑥) ⊆ ( bday ‘𝑥)
31		simpr 488	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( bday ‘𝑥) ∈ 𝑎)
32		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ No )
33	31, 32	jca 515	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → (( bday ‘𝑥) ∈ 𝑎 ∧ 𝑥 ∈ No ))
34		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
35		sseq2 3920	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ( bday ‘𝑥) → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ ( bday ‘𝑥)))
36		fveq2 6662	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ( bday ‘𝑥) → ( M ‘𝑏) = ( M ‘( bday ‘𝑥)))
37	36	eleq2d 2837	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ( bday ‘𝑥) → (𝑦 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘( bday ‘𝑥))))
38	35, 37	imbi12d 348	. . . . . . . . . . . . 13 ⊢ (𝑏 = ( bday ‘𝑥) → ((( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ↔ (( bday ‘𝑦) ⊆ ( bday ‘𝑥) → 𝑦 ∈ ( M ‘( bday ‘𝑥)))))
39		fveq2 6662	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ( bday ‘𝑦) = ( bday ‘𝑥))
40	39	sseq1d 3925	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (( bday ‘𝑦) ⊆ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ⊆ ( bday ‘𝑥)))
41		eleq1 2839	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝑥 ∈ ( M ‘( bday ‘𝑥))))
42	40, 41	imbi12d 348	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((( bday ‘𝑦) ⊆ ( bday ‘𝑥) → 𝑦 ∈ ( M ‘( bday ‘𝑥))) ↔ (( bday ‘𝑥) ⊆ ( bday ‘𝑥) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))))
43	38, 42	rspc2v 3553	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑥) ∈ 𝑎 ∧ 𝑥 ∈ No ) → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → (( bday ‘𝑥) ⊆ ( bday ‘𝑥) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))))
44	33, 34, 43	sylc 65	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → (( bday ‘𝑥) ⊆ ( bday ‘𝑥) → 𝑥 ∈ ( M ‘( bday ‘𝑥))))
45	30, 44	mpi 20	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))
46	29, 45	sseldd 3895	. . . . . . . . 9 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ ( M ‘𝑎))
47	46	ex 416	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ∈ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
48		madebdaylemlrcut 33662	. . . . . . . . . . . 12 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
49	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( bday ‘𝑥) ∈ On)
50		lltropt 33638	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) <<s ( R ‘𝑥))
51		leftssold 33644	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
52		rightssold 33645	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
53		madecut 33648	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑥) ∈ On ∧ ( L ‘𝑥) <<s ( R ‘𝑥)) ∧ (( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)) ∧ ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))) → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday ‘𝑥)))
54	49, 50, 51, 52, 53	syl22anc 837	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday ‘𝑥)))
55	54	adantl 485	. . . . . . . . . . . 12 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday ‘𝑥)))
56	48, 55	eqeltrrd 2853	. . . . . . . . . . 11 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))
57		raleq 3323	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑥) = 𝑎 → (∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ↔ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))))
58	57	anbi1d 632	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑥) = 𝑎 → ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) ↔ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No )))
59		fveq2 6662	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑥) = 𝑎 → ( M ‘( bday ‘𝑥)) = ( M ‘𝑎))
60	59	eleq2d 2837	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑥) = 𝑎 → (𝑥 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝑥 ∈ ( M ‘𝑎)))
61	58, 60	imbi12d 348	. . . . . . . . . . 11 ⊢ (( bday ‘𝑥) = 𝑎 → (((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘( bday ‘𝑥))) ↔ ((∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘𝑎))))
62	56, 61	mpbii 236	. . . . . . . . . 10 ⊢ (( bday ‘𝑥) = 𝑎 → ((∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘𝑎)))
63	62	com12 32	. . . . . . . . 9 ⊢ ((∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) = 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
64	63	adantll 713	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) = 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
65	47, 64	jaod 856	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → ((( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎) → 𝑥 ∈ ( M ‘𝑎)))
66	23, 65	sylbid 243	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
67	66	ralrimiva 3113	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
68	67	ex 416	. . . 4 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎))))
69	14, 19, 68	tfis3 7576	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)))
70		fveq2 6662	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
71	70	sseq1d 3925	. . . . 5 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) ⊆ 𝐴 ↔ ( bday ‘𝑋) ⊆ 𝐴))
72		eleq1 2839	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ( M ‘𝐴) ↔ 𝑋 ∈ ( M ‘𝐴)))
73	71, 72	imbi12d 348	. . . 4 ⊢ (𝑥 = 𝑋 → ((( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)) ↔ (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴))))
74	73	rspccva 3542	. . 3 ⊢ ((∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴)))
75	69, 74	sylan 583	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴)))
76	3, 75	impbid 215	1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3860   class class class wbr 5035  Oncon0 6173  ‘cfv 6339  (class class class)co 7155   No csur 33432   bday cbday 33434   <<s csslt 33564   |s cscut 33566   M cmade 33612   O cold 33613   L cleft 33615   R cright 33616

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 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-wrecs 7962  df-recs 8023  df-1o 8117  df-2o 8118  df-no 33435  df-slt 33436  df-bday 33437  df-sslt 33565  df-scut 33567  df-made 33617  df-old 33618  df-left 33620  df-right 33621

This theorem is referenced by:  oldbday  33664  newbday  33665  lrcut  33666