Theorem madebday 27872

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 27860	. 2 ⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)
2		sseq2 4003	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑏))
3		fveq2 6896	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
4	3	eleq2d 2811	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝑏)))
5	2, 4	imbi12d 343	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏))))
6	5	ralbidv 3167	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏))))
7		fveq2 6896	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
8	7	sseq1d 4008	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑥) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ 𝑏))
9		eleq1 2813	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘𝑏)))
10	8, 9	imbi12d 343	. . . . . 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 286	. . . 4 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))))
13		sseq2 4003	. . . . . 6 ⊢ (𝑎 = 𝐴 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝐴))
14		fveq2 6896	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
15	14	eleq2d 2811	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝐴)))
16	13, 15	imbi12d 343	. . . . 5 ⊢ (𝑎 = 𝐴 → ((( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴))))
17	16	ralbidv 3167	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴))))
18		bdayelon 27755	. . . . . . . . 9 ⊢ ( bday ‘𝑥) ∈ On
19		onsseleq 6412	. . . . . . . . 9 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝑎 ∈ On) → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
20	18, 19	mpan 688	. . . . . . . 8 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
21	20	ad2antrr 724	. . . . . . 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 765	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → 𝑎 ∈ On)
23		onelss 6413	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ∈ 𝑎 → ( bday ‘𝑥) ⊆ 𝑎))
24	23	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ∈ 𝑎 → ( bday ‘𝑥) ⊆ 𝑎))
25	24	imp 405	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( bday ‘𝑥) ⊆ 𝑎)
26		madess 27849	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ On ∧ ( bday ‘𝑥) ⊆ 𝑎) → ( M ‘( bday ‘𝑥)) ⊆ ( M ‘𝑎))
27	22, 25, 26	syl2an2r 683	. . . . . . . . . 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 3999	. . . . . . . . . . 11 ⊢ ( bday ‘𝑥) ⊆ ( bday ‘𝑥)
29		simpr 483	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( bday ‘𝑥) ∈ 𝑎)
30		simplr 767	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ No )
31	29, 30	jca 510	. . . . . . . . . . . 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 774	. . . . . . . . . . . 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 4003	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ( bday ‘𝑥) → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ ( bday ‘𝑥)))
34		fveq2 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ( bday ‘𝑥) → ( M ‘𝑏) = ( M ‘( bday ‘𝑥)))
35	34	eleq2d 2811	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ( bday ‘𝑥) → (𝑦 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘( bday ‘𝑥))))
36	33, 35	imbi12d 343	. . . . . . . . . . . . 13 ⊢ (𝑏 = ( bday ‘𝑥) → ((( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ↔ (( bday ‘𝑦) ⊆ ( bday ‘𝑥) → 𝑦 ∈ ( M ‘( bday ‘𝑥)))))
37		fveq2 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ( bday ‘𝑦) = ( bday ‘𝑥))
38	37	sseq1d 4008	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (( bday ‘𝑦) ⊆ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ⊆ ( bday ‘𝑥)))
39		eleq1 2813	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝑥 ∈ ( M ‘( bday ‘𝑥))))
40	38, 39	imbi12d 343	. . . . . . . . . . . . 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 3977	. . . . . . . . 9 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ ( M ‘𝑎))
45	44	ex 411	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ∈ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
46		madebdaylemlrcut 27871	. . . . . . . . . . . 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 27845	. . . . . . . . . . . . . . 15 ⊢ ( L ‘𝑥) <<s ( R ‘𝑥)
49	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) <<s ( R ‘𝑥))
50		leftssold 27851	. . . . . . . . . . . . . . 15 ⊢ ( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥))
51	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
52		rightssold 27852	. . . . . . . . . . . . . . 15 ⊢ ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥))
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
54		madecut 27855	. . . . . . . . . . . . . 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 837	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday ‘𝑥)))
56	55	adantl 480	. . . . . . . . . . . 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 2826	. . . . . . . . . . 11 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))
58		raleq 3311	. . . . . . . . . . . . 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 629	. . . . . . . . . . . 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 6896	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑥) = 𝑎 → ( M ‘( bday ‘𝑥)) = ( M ‘𝑎))
61	60	eleq2d 2811	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑥) = 𝑎 → (𝑥 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝑥 ∈ ( M ‘𝑎)))
62	59, 61	imbi12d 343	. . . . . . . . . . 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 232	. . . . . . . . . 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 712	. . . . . . . 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 857	. . . . . . 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 239	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
68	67	ralrimiva 3135	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
69	68	ex 411	. . . 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 7863	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)))
71		fveq2 6896	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
72	71	sseq1d 4008	. . . . 5 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) ⊆ 𝐴 ↔ ( bday ‘𝑋) ⊆ 𝐴))
73		eleq1 2813	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ( M ‘𝐴) ↔ 𝑋 ∈ ( M ‘𝐴)))
74	72, 73	imbi12d 343	. . . 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 578	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴)))
77	1, 76	impbid2 225	1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  ∀wral 3050   ⊆ wss 3944   class class class wbr 5149  Oncon0 6371  ‘cfv 6549  (class class class)co 7419   No csur 27618   bday cbday 27620   <<s csslt 27759   |s cscut 27761   M cmade 27815   O cold 27816   L cleft 27818   R cright 27819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-1o 8487  df-2o 8488  df-no 27621  df-slt 27622  df-bday 27623  df-sslt 27760  df-scut 27762  df-made 27820  df-old 27821  df-left 27823  df-right 27824

This theorem is referenced by:  oldbday  27873  newbday  27874  lrcut  27875  sltonold  28203