Theorem madebday 34007

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 33997	. 2 ⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)
2		sseq2 3943	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑏))
3		fveq2 6756	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
4	3	eleq2d 2824	. . . . . . 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 3120	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏))))
7		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
8	7	sseq1d 3948	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑥) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ 𝑏))
9		eleq1 2826	. . . . . . 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 3372	. . . . 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 3943	. . . . . 6 ⊢ (𝑎 = 𝐴 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝐴))
14		fveq2 6756	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
15	14	eleq2d 2824	. . . . . 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 3120	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴))))
18		bdayelon 33898	. . . . . . . . 9 ⊢ ( bday ‘𝑥) ∈ On
19		onsseleq 6292	. . . . . . . . 9 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝑎 ∈ On) → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
20	18, 19	mpan 686	. . . . . . . 8 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
21	20	ad2antrr 722	. . . . . . 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 763	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → 𝑎 ∈ On)
23		onelss 6293	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ∈ 𝑎 → ( bday ‘𝑥) ⊆ 𝑎))
24	23	ad2antrr 722	. . . . . . . . . . . 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 33986	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ On ∧ ( bday ‘𝑥) ⊆ 𝑎) → ( M ‘( bday ‘𝑥)) ⊆ ( M ‘𝑎))
27	22, 25, 26	syl2an2r 681	. . . . . . . . . 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 3939	. . . . . . . . . . 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 765	. . . . . . . . . . . . 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 772	. . . . . . . . . . . 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 3943	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ( bday ‘𝑥) → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ ( bday ‘𝑥)))
34		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ( bday ‘𝑥) → ( M ‘𝑏) = ( M ‘( bday ‘𝑥)))
35	34	eleq2d 2824	. . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ( bday ‘𝑦) = ( bday ‘𝑥))
38	37	sseq1d 3948	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (( bday ‘𝑦) ⊆ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ⊆ ( bday ‘𝑥)))
39		eleq1 2826	. . . . . . . . . . . . . 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 3562	. . . . . . . . . . . 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 3918	. . . . . . . . 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 34006	. . . . . . . . . . . 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 33983	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) <<s ( R ‘𝑥))
49		leftssold 33988	. . . . . . . . . . . . . . 15 ⊢ ( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥))
50	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
51		rightssold 33989	. . . . . . . . . . . . . . 15 ⊢ ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥))
52	51	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
53		madecut 33992	. . . . . . . . . . . . . 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	47, 48, 50, 52, 53	syl22anc 835	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday ‘𝑥)))
55	54	adantl 481	. . . . . . . . . . . 12 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday ‘𝑥)))
56	46, 55	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))
57		raleq 3333	. . . . . . . . . . . . 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 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 )))
59		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑥) = 𝑎 → ( M ‘( bday ‘𝑥)) = ( M ‘𝑎))
60	59	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑥) = 𝑎 → (𝑥 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝑥 ∈ ( M ‘𝑎)))
61	58, 60	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))))
62	56, 61	mpbii 232	. . . . . . . . . 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 710	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) = 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
65	45, 64	jaod 855	. . . . . . 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	21, 65	sylbid 239	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
67	66	ralrimiva 3107	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
68	67	ex 412	. . . 4 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎))))
69	12, 17, 68	tfis3 7679	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)))
70		fveq2 6756	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
71	70	sseq1d 3948	. . . . 5 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) ⊆ 𝐴 ↔ ( bday ‘𝑋) ⊆ 𝐴))
72		eleq1 2826	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ( M ‘𝐴) ↔ 𝑋 ∈ ( M ‘𝐴)))
73	71, 72	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑋 → ((( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)) ↔ (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴))))
74	73	rspccva 3551	. . 3 ⊢ ((∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴)))
75	69, 74	sylan 579	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴)))
76	1, 75	impbid2 225	1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883   class class class wbr 5070  Oncon0 6251  ‘cfv 6418  (class class class)co 7255   No csur 33770   bday cbday 33772   <<s csslt 33902   |s cscut 33904   M cmade 33953   O cold 33954   L cleft 33956   R cright 33957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959  df-left 33961  df-right 33962

This theorem is referenced by:  oldbday  34008  newbday  34009  lrcut  34010