Theorem nnawordex 8646

Description: Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnawordex	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnawordex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7428	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
2	1	sseq2d 3940	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
3		simplr 781	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ ω)
4		nnon 7883	. . . . . . . 8 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
5	3, 4	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On)
6		simpll 779	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ ω)
7		nnaword2 8639	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +o 𝐵))
8	3, 6, 7	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ (𝐴 +o 𝐵))
9	2, 5, 8	elrabd 3646	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
10		intss1 4922	. . . . . 6 ⊢ (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵)
11	9, 10	syl 18	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵)
12		ssrab2 4005	. . . . . . . 8 ⊢ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ On
13	9	ne0d 4287	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ≠ ∅)
14		oninton 7809	. . . . . . . 8 ⊢ (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On)
15	12, 13, 14	sylancr 599	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On)
16		eloni 6372	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On → Ord ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
17	15, 16	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → Ord ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
18		ordom 7887	. . . . . 6 ⊢ Ord ω
19		ordtr2 6408	. . . . . 6 ⊢ ((Ord ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∧ Ord ω) → ((∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵 ∧ 𝐵 ∈ ω) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω))
20	17, 18, 19	sylancl 598	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ((∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵 ∧ 𝐵 ∈ ω) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω))
21	11, 3, 20	mp2and 712	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω)
22		nna0 8613	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
23	22	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∅) = 𝐴)
24		simpr 490	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
25	23, 24	eqsstrd 3942	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∅) ⊆ 𝐵)
26		oveq2 7428	. . . . . . . 8 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = (𝐴 +o ∅))
27	26	sseq1d 3939	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → ((𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵 ↔ (𝐴 +o ∅) ⊆ 𝐵))
28	25, 27	syl5ibrcom 250	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵))
29		simprr 785	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)
30	29	oveq2d 7436	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = (𝐴 +o suc 𝑥))
31	6	adantr 486	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → 𝐴 ∈ ω)
32		simprl 783	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → 𝑥 ∈ ω)
33		nnasuc 8615	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
34	31, 32, 33	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
35	30, 34	eqtrd 2795	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = suc (𝐴 +o 𝑥))
36		nnord 7885	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → Ord 𝐵)
37	3, 36	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → Ord 𝐵)
38	37	adantr 486	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → Ord 𝐵)
39		nnon 7883	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
40	39	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → 𝑥 ∈ On)
41		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
42	41	sucid 6447	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ suc 𝑥
43		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)
44	42, 43	eleqtrrid 2867	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → 𝑥 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
45		oveq2 7428	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o 𝑥))
46	45	sseq2d 3940	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑥)))
47	46	onnminsb 7813	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑥 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +o 𝑥)))
48	40, 44, 47	sylc 66	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → ¬ 𝐵 ⊆ (𝐴 +o 𝑥))
49	48	adantl 487	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → ¬ 𝐵 ⊆ (𝐴 +o 𝑥))
50		nnacl 8620	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
51	31, 32, 50	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o 𝑥) ∈ ω)
52		nnord 7885	. . . . . . . . . . . . 13 ⊢ ((𝐴 +o 𝑥) ∈ ω → Ord (𝐴 +o 𝑥))
53	51, 52	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → Ord (𝐴 +o 𝑥))
54		ordtri1 6396	. . . . . . . . . . . 12 ⊢ ((Ord 𝐵 ∧ Ord (𝐴 +o 𝑥)) → (𝐵 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐵))
55	38, 53, 54	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐵 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐵))
56	55	con2bid 357	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → ((𝐴 +o 𝑥) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +o 𝑥)))
57	49, 56	mpbird 260	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o 𝑥) ∈ 𝐵)
58		ordsucss 7829	. . . . . . . . 9 ⊢ (Ord 𝐵 → ((𝐴 +o 𝑥) ∈ 𝐵 → suc (𝐴 +o 𝑥) ⊆ 𝐵))
59	38, 57, 58	sylc 66	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → suc (𝐴 +o 𝑥) ⊆ 𝐵)
60	35, 59	eqsstrd 3942	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵)
61	60	rexlimdvaa 3164	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (∃𝑥 ∈ ω ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥 → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵))
62		nn0suc 7906	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω → (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ ∨ ∃𝑥 ∈ ω ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥))
63	21, 62	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ ∨ ∃𝑥 ∈ ω ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥))
64	28, 61, 63	mpjaod 874	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵)
65		onint 7804	. . . . . . 7 ⊢ (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
66	12, 13, 65	sylancr 599	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
67		nfrab1 3431	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
68	67	nfint 4916	. . . . . . . 8 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
69		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑦On
70		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐵
71		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
72		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑦 +o
73	71, 72, 68	nfov 7450	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
74	70, 73	nfss 3901	. . . . . . . 8 ⊢ Ⅎ𝑦 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
75		oveq2 7428	. . . . . . . . 9 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
76	75	sseq2d 3940	. . . . . . . 8 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})))
77	68, 69, 74, 76	elrabf 3641	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ↔ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On ∧ 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})))
78	77	simprbi 503	. . . . . 6 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
79	66, 78	syl 18	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
80	64, 79	eqssd 3925	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = 𝐵)
81		oveq2 7428	. . . . . 6 ⊢ (𝑥 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑥) = (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
82	81	eqeq1d 2762	. . . . 5 ⊢ (𝑥 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = 𝐵))
83	82	rspcev 3576	. . . 4 ⊢ ((∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω ∧ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
84	21, 80, 83	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
85	84	ex 418	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
86		nnaword1 8638	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
87	86	adantlr 728	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
88		sseq2 3934	. . . 4 ⊢ ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ 𝐴 ⊆ 𝐵))
89	87, 88	syl5ibcom 248	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
90	89	rexlimdva 3163	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
91	85, 90	impbid 215	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  {crab 3412   ⊆ wss 3895  ∅c0 4278  ∩ cint 4906  Ord word 6361  Oncon0 6362  suc csuc 6364  (class class class)co 7420  ωcom 7877   +_o coa 8473

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-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-oadd 8480

This theorem is used by:  nnaordex  8647  eldifsucnn  8673  unfilem1  9297  ttrcltr  9717  hashdom  14523  precsexlem6  28598  precsexlem7  28599