Theorem madebday 28295

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 28283	. 2 ⊢ (𝑋 ∈ (M‘𝐴) → (bday‘𝑋) ⊆ 𝐴)
2		sseq2 3935	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ((bday‘𝑥) ⊆ 𝑎 ↔ (bday‘𝑥) ⊆ 𝑏))
3		fveq2 6890	. . . . . . . 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 6890	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (bday‘𝑥) = (bday‘𝑦))
8	7	sseq1d 3940	. . . . . . 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 3935	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((bday‘𝑥) ⊆ 𝑎 ↔ (bday‘𝑥) ⊆ 𝐴))
14		fveq2 6890	. . . . . . 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 28147	. . . . . . . . 9 ⊢ (bday‘𝑥) ∈ On
19		onsseleq 6407	. . . . . . . . 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 6408	. . . . . . . . . . . . 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 28261	. . . . . . . . . . 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 3931	. . . . . . . . . . 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 3935	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (bday‘𝑥) → ((bday‘𝑦) ⊆ 𝑏 ↔ (bday‘𝑦) ⊆ (bday‘𝑥)))
34		fveq2 6890	. . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (bday‘𝑦) = (bday‘𝑥))
38	37	sseq1d 3940	. . . . . . . . . . . . . 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 3586	. . . . . . . . . . . 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 3910	. . . . . . . . 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 28294	. . . . . . . . . . . 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 28257	. . . . . . . . . . . . . . 15 ⊢ (L‘𝑥) <<s (R‘𝑥)
49	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → (L‘𝑥) <<s (R‘𝑥))
50		leftssold 28266	. . . . . . . . . . . . . . 15 ⊢ (L‘𝑥) ⊆ (O‘(bday‘𝑥))
51	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → (L‘𝑥) ⊆ (O‘(bday‘𝑥)))
52		rightssold 28267	. . . . . . . . . . . . . . 15 ⊢ (R‘𝑥) ⊆ (O‘(bday‘𝑥))
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ No → (R‘𝑥) ⊆ (O‘(bday‘𝑥)))
54		madecut 28278	. . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . 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 7873	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ No ((bday‘𝑥) ⊆ 𝐴 → 𝑥 ∈ (M‘𝐴)))
71		fveq2 6890	. . . . . 6 ⊢ (𝑥 = 𝑋 → (bday‘𝑥) = (bday‘𝑋))
72	71	sseq1d 3940	. . . . 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 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3896   class class class wbr 5102  Oncon0 6365  ‘cfv 6543  (class class class)co 7424  Nocsur 28006  bdaycbday 28008   <<s cslts 28152   |s ccuts 28154  Mcmade 28217  Ocold 28218  Lcleft 28220  Rcright 28221

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-1o 8484  df-2o 8485  df-no 28009  df-lts 28010  df-bday 28011  df-slts 28153  df-cuts 28155  df-made 28222  df-old 28223  df-left 28225  df-right 28226

This theorem is used by:  oldbday  28296  newbday  28297  lrcut  28299  ltonold  28656  onsbnd2  28677  bdayfinbndlem1  28862