Theorem madebday 28166

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 28154	. 2 ⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)
2		sseq2 3957	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑏))
3		fveq2 6879	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
4	3	eleq2d 2846	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝑏)))
5	2, 4	imbi12d 347	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏))))
6	5	ralbidv 3185	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏))))
7		fveq2 6879	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
8	7	sseq1d 3962	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑥) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ 𝑏))
9		eleq1 2848	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘𝑏)))
10	8, 9	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏)) ↔ (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))))
11	10	cbvralvw 3240	. . . . 5 ⊢ (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑏 → 𝑥 ∈ ( M ‘𝑏)) ↔ ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
12	6, 11	bitrdi 290	. . . 4 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))))
13		sseq2 3957	. . . . . 6 ⊢ (𝑎 = 𝐴 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝐴))
14		fveq2 6879	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
15	14	eleq2d 2846	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝐴)))
16	13, 15	imbi12d 347	. . . . 5 ⊢ (𝑎 = 𝐴 → ((( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴))))
17	16	ralbidv 3185	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴))))
18		bdayon 28018	. . . . . . . . 9 ⊢ ( bday ‘𝑥) ∈ On
19		onsseleq 6399	. . . . . . . . 9 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝑎 ∈ On) → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
20	18, 19	mpan 703	. . . . . . . 8 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ⊆ 𝑎 ↔ (( bday ‘𝑥) ∈ 𝑎 ∨ ( bday ‘𝑥) = 𝑎)))
21	20	ad2antrr 739	. . . . . . 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 779	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → 𝑎 ∈ On)
23		onelss 6400	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ On → (( bday ‘𝑥) ∈ 𝑎 → ( bday ‘𝑥) ⊆ 𝑎))
24	23	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ∈ 𝑎 → ( bday ‘𝑥) ⊆ 𝑎))
25	24	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( bday ‘𝑥) ⊆ 𝑎)
26		madess 28132	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ On ∧ ( bday ‘𝑥) ⊆ 𝑎) → ( M ‘( bday ‘𝑥)) ⊆ ( M ‘𝑎))
27	22, 25, 26	syl2an2r 698	. . . . . . . . . 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 3953	. . . . . . . . . . 11 ⊢ ( bday ‘𝑥) ⊆ ( bday ‘𝑥)
29		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → ( bday ‘𝑥) ∈ 𝑎)
30		simplr 781	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ No )
31	29, 30	jca 521	. . . . . . . . . . . 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 788	. . . . . . . . . . . 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 3957	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ( bday ‘𝑥) → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ ( bday ‘𝑥)))
34		fveq2 6879	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ( bday ‘𝑥) → ( M ‘𝑏) = ( M ‘( bday ‘𝑥)))
35	34	eleq2d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ( bday ‘𝑥) → (𝑦 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘( bday ‘𝑥))))
36	33, 35	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑏 = ( bday ‘𝑥) → ((( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ↔ (( bday ‘𝑦) ⊆ ( bday ‘𝑥) → 𝑦 ∈ ( M ‘( bday ‘𝑥)))))
37		fveq2 6879	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ( bday ‘𝑦) = ( bday ‘𝑥))
38	37	sseq1d 3962	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (( bday ‘𝑦) ⊆ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ⊆ ( bday ‘𝑥)))
39		eleq1 2848	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝑥 ∈ ( M ‘( bday ‘𝑥))))
40	38, 39	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((( bday ‘𝑦) ⊆ ( bday ‘𝑥) → 𝑦 ∈ ( M ‘( bday ‘𝑥))) ↔ (( bday ‘𝑥) ⊆ ( bday ‘𝑥) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))))
41	36, 40	rspc2v 3587	. . . . . . . . . . . 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 66	. . . . . . . . . . 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 21	. . . . . . . . . 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 3932	. . . . . . . . 9 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) ∧ ( bday ‘𝑥) ∈ 𝑎) → 𝑥 ∈ ( M ‘𝑎))
45	44	ex 418	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ∈ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
46		madebdaylemlrcut 28165	. . . . . . . . . . . 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		lltr 28128	. . . . . . . . . . . . . . 15 ⊢ ( L ‘𝑥) <<s ( R ‘𝑥)
49	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) <<s ( R ‘𝑥))
50		leftssold 28137	. . . . . . . . . . . . . . 15 ⊢ ( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥))
51	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( L ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
52		rightssold 28138	. . . . . . . . . . . . . . 15 ⊢ ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥))
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → ( R ‘𝑥) ⊆ ( O ‘( bday ‘𝑥)))
54		madecut 28149	. . . . . . . . . . . . . 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 852	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday ‘𝑥)))
56	55	adantl 487	. . . . . . . . . . . 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 2861	. . . . . . . . . . 11 ⊢ ((∀𝑏 ∈ ( bday ‘𝑥)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘( bday ‘𝑥)))
58		raleq 3316	. . . . . . . . . . . . 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 643	. . . . . . . . . . . 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 6879	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑥) = 𝑎 → ( M ‘( bday ‘𝑥)) = ( M ‘𝑎))
61	60	eleq2d 2846	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑥) = 𝑎 → (𝑥 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝑥 ∈ ( M ‘𝑎)))
62	59, 61	imbi12d 347	. . . . . . . . . . 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 236	. . . . . . . . . 10 ⊢ (( bday ‘𝑥) = 𝑎 → ((∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘𝑎)))
64	63	com12 33	. . . . . . . . 9 ⊢ ((∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) = 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
65	64	adantll 727	. . . . . . . 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 873	. . . . . . 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 243	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 ∈ No ) → (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
68	67	ralrimiva 3154	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝑎 → 𝑥 ∈ ( M ‘𝑎)))
69	68	ex 418	. . . 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 7855	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)))
71		fveq2 6879	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
72	71	sseq1d 3962	. . . . 5 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) ⊆ 𝐴 ↔ ( bday ‘𝑋) ⊆ 𝐴))
73		eleq1 2848	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ( M ‘𝐴) ↔ 𝑋 ∈ ( M ‘𝐴)))
74	72, 73	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑋 → ((( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)) ↔ (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴))))
75	74	rspccva 3575	. . 3 ⊢ ((∀𝑥 ∈ No (( bday ‘𝑥) ⊆ 𝐴 → 𝑥 ∈ ( M ‘𝐴)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴)))
76	70, 75	sylan 592	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ⊆ 𝐴 → 𝑋 ∈ ( M ‘𝐴)))
77	1, 76	impbid2 229	1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899   class class class wbr 5103  Oncon0 6357  ‘cfv 6533  (class class class)co 7414   No csur 27877   bday cbday 27879   <<s cslts 28023   |s ccuts 28025   M cmade 28088   O cold 28089   L cleft 28091   R cright 28092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-1o 8456  df-2o 8457  df-no 27880  df-lts 27881  df-bday 27882  df-slts 28024  df-cuts 28026  df-made 28093  df-old 28094  df-left 28096  df-right 28097

This theorem is used by:  oldbday  28167  newbday  28168  lrcut  28170  ltonold  28527  onsbnd2  28548  bdayfinbndlem1  28733