Theorem madebday 27868

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 27856	. 2 ⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)
2		sseq2 3990	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑏))
3		fveq2 6881	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
4	3	eleq2d 2821	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝑏)))
5	2, 4	imbi12d 344	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏))))
6	5	ralbidv 3164	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏))))
7		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
8	7	sseq1d 3995	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑥) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ 𝑏))
9		eleq1 2823	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘𝑏)))
10	8, 9	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏)) ↔ (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))))
11	10	cbvralvw 3224	. . . . 5 ⊢ (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏)) ↔ ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
12	6, 11	bitrdi 287	. . . 4 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))))
13		sseq2 3990	. . . . . 6 ⊢ (𝑎 = 𝐴 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝐴))
14		fveq2 6881	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
15	14	eleq2d 2821	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝐴)))
16	13, 15	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝐴 → ((( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴))))
17	16	ralbidv 3164	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴))))
18		bdayelon 27745	. . . . . . . . 9 ⊢ ( bday ‘𝑥) ∈ On
19		onsseleq 6398	. . . . . . . . 9 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝑎 ∈ On) → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
20	18, 19	mpan 690	. . . . . . . 8 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
21	20	ad2antrr 726	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
22		simpll 766	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → 𝑎 ∈ On)
23		onelss 6399	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ∈ 𝑎 → ( bday ‘𝑥) ⊆ 𝑎))
24	23	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ∈ 𝑎 → ( bday ‘𝑥) ⊆ 𝑎))
25	24	imp 406	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( bday ‘𝑥) ⊆ 𝑎)
26		madess 27845	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ On ∧ ( bday ‘𝑥) ⊆ 𝑎) → ( M ‘( bday ‘𝑥)) ⊆ ( M ‘𝑎))
27	22, 25, 26	syl2an2r 685	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( M ‘( bday ‘𝑥)) ⊆ ( M ‘𝑎))
28		ssid 3986	. . . . . . . . . . 11 ⊢ ( bday ‘𝑥) ⊆ ( bday ‘𝑥)
29		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( bday ‘𝑥) ∈ 𝑎)
30		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ No )
31	29, 30	jca 511	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → (( bday ‘𝑥) ∈ 𝑎 ∧ 𝑥 ∈ No ))
32		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)))
33		sseq2 3990	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ( bday ‘𝑥) → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ ( bday ‘𝑥)))
34		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ( bday ‘𝑥) → ( M ‘𝑏) = ( M ‘( bday ‘𝑥)))
35	34	eleq2d 2821	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ( bday ‘𝑥) → (𝑦 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘( bday ‘𝑥))))
36	33, 35	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑏 = ( bday ‘𝑥) → ((( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ↔ (( bday ‘𝑦) ⊆ ( bday ‘𝑥) → 𝑦 ∈ ( M ‘( bday ‘𝑥)))))
37		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ( bday ‘𝑦) = ( bday ‘𝑥))
38	37	sseq1d 3995	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (( bday ‘𝑦) ⊆ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ⊆ ( bday ‘𝑥)))
39		eleq1 2823	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝑥 ∈ ( M ‘( bday ‘𝑥))))
40	38, 39	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((( bday ‘𝑦) ⊆ ( bday ‘𝑥) → 𝑦 ∈ ( M ‘( bday ‘𝑥))) ↔ (( bday ‘𝑥) ⊆ ( bday ‘𝑥) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))))
41	36, 40	rspc2v 3617	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑥) ∈ 𝑎 ∧ 𝑥 ∈ No ) → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → (( bday ‘𝑥) ⊆ ( bday ‘𝑥) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))))
42	31, 32, 41	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))))
43	28, 42	mpi 20	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))
44	27, 43	sseldd 3964	. . . . . . . . 9 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ ( M ‘𝑎))
45	44	ex 412	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ∈ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
46		madebdaylemlrcut 27867	. . . . . . . . . . . 12 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
47	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( bday ‘𝑥) ∈ On)
48		lltropt 27841	. . . . . . . . . . . . . . 15 ⊢ ( L ‘𝑥) <<s ( R ‘𝑥)
49	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) <<s ( R ‘𝑥))
50		leftssold 27847	. . . . . . . . . . . . . . 15 ⊢ ( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥))
51	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
52		rightssold 27848	. . . . . . . . . . . . . . 15 ⊢ ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥))
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
54		madecut 27851	. . . . . . . . . . . . . 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)))
55	47, 49, 51, 53, 54	syl22anc 838	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday ‘𝑥)))
56	55	adantl 481	. . . . . . . . . . . 12 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday ‘𝑥)))
57	46, 56	eqeltrrd 2836	. . . . . . . . . . 11 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))
58		raleq 3306	. . . . . . . . . . . . 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))))
59	58	anbi1d 631	. . . . . . . . . . . 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 )))
60		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑥) = 𝑎 → ( M ‘( bday ‘𝑥)) = ( M ‘𝑎))
61	60	eleq2d 2821	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑥) = 𝑎 → (𝑥 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝑥 ∈ ( M ‘𝑎)))
62	59, 61	imbi12d 344	. . . . . . . . . . 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))))
63	57, 62	mpbii 233	. . . . . . . . . 10 ⊢ (( bday ‘𝑥) = 𝑎 → ((∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘𝑎)))
64	63	com12 32	. . . . . . . . 9 ⊢ ((∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) = 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
65	64	adantll 714	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) = 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
66	45, 65	jaod 859	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → ((( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎) → 𝑥 ∈ ( M ‘𝑎)))
67	21, 66	sylbid 240	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
68	67	ralrimiva 3133	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
69	68	ex 412	. . . 4 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎))))
70	12, 17, 69	tfis3 7858	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)))
71		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
72	71	sseq1d 3995	. . . . 5 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) ⊆ 𝐴 ↔ ( bday ‘𝑋) ⊆ 𝐴))
73		eleq1 2823	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ( M ‘𝐴) ↔ 𝑋 ∈ ( M ‘𝐴)))
74	72, 73	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑋 → ((( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)) ↔ (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴))))
75	74	rspccva 3605	. . 3 ⊢ ((∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴)))
76	70, 75	sylan 580	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴)))
77	1, 76	impbid2 226	1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ⊆ wss 3931   class class class wbr 5124  Oncon0 6357  ‘cfv 6536  (class class class)co 7410   No csur 27608   bday cbday 27610   <<s csslt 27749   |s cscut 27751   M cmade 27807   O cold 27808   L cleft 27810   R cright 27811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612  df-bday 27613  df-sslt 27750  df-scut 27752  df-made 27812  df-old 27813  df-left 27815  df-right 27816

This theorem is referenced by:  oldbday  27869  newbday  27870  lrcut  27872  sltonold  28219