Theorem nnawordex 8565

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 7366	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
2	1	sseq2d 3966	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
3		simplr 768	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ ω)
4		nnon 7814	. . . . . . . 8 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
5	3, 4	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On)
6		simpll 766	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ ω)
7		nnaword2 8558	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +o 𝐵))
8	3, 6, 7	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ (𝐴 +o 𝐵))
9	2, 5, 8	elrabd 3648	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
10		intss1 4918	. . . . . 6 ⊢ (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵)
11	9, 10	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵)
12		ssrab2 4032	. . . . . . . 8 ⊢ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ On
13	9	ne0d 4294	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ≠ ∅)
14		oninton 7740	. . . . . . . 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 587	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On)
16		eloni 6327	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On → Ord ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
17	15, 16	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → Ord ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
18		ordom 7818	. . . . . 6 ⊢ Ord ω
19		ordtr2 6362	. . . . . 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 586	. . . . 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 699	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω)
22		nna0 8532	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
23	22	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∅) = 𝐴)
24		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
25	23, 24	eqsstrd 3968	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∅) ⊆ 𝐵)
26		oveq2 7366	. . . . . . . 8 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = (𝐴 +o ∅))
27	26	sseq1d 3965	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → ((𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵 ↔ (𝐴 +o ∅) ⊆ 𝐵))
28	25, 27	syl5ibrcom 247	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵))
29		simprr 772	. . . . . . . . . 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 7374	. . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → 𝐴 ∈ ω)
32		simprl 770	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → 𝑥 ∈ ω)
33		nnasuc 8534	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
34	31, 32, 33	syl2anc 584	. . . . . . . . 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 2771	. . . . . . . 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 7816	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → Ord 𝐵)
37	3, 36	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → Ord 𝐵)
38	37	adantr 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → Ord 𝐵)
39		nnon 7814	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
40	39	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → 𝑥 ∈ On)
41		vex 3444	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
42	41	sucid 6401	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ suc 𝑥
43		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)
44	42, 43	eleqtrrid 2843	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → 𝑥 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
45		oveq2 7366	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o 𝑥))
46	45	sseq2d 3966	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑥)))
47	46	onnminsb 7744	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑥 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +o 𝑥)))
48	40, 44, 47	sylc 65	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → ¬ 𝐵 ⊆ (𝐴 +o 𝑥))
49	48	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → ¬ 𝐵 ⊆ (𝐴 +o 𝑥))
50		nnacl 8539	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
51	31, 32, 50	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o 𝑥) ∈ ω)
52		nnord 7816	. . . . . . . . . . . . 13 ⊢ ((𝐴 +o 𝑥) ∈ ω → Ord (𝐴 +o 𝑥))
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → Ord (𝐴 +o 𝑥))
54		ordtri1 6350	. . . . . . . . . . . 12 ⊢ ((Ord 𝐵 ∧ Ord (𝐴 +o 𝑥)) → (𝐵 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐵))
55	38, 53, 54	syl2anc 584	. . . . . . . . . . 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 354	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o 𝑥) ∈ 𝐵)
58		ordsucss 7760	. . . . . . . . 9 ⊢ (Ord 𝐵 → ((𝐴 +o 𝑥) ∈ 𝐵 → suc (𝐴 +o 𝑥) ⊆ 𝐵))
59	38, 57, 58	sylc 65	. . . . . . . 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 3968	. . . . . . 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 3138	. . . . . 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 7836	. . . . . . 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 17	. . . . . 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 860	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵)
65		onint 7735	. . . . . . 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 587	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
67		nfrab1 3419	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
68	67	nfint 4912	. . . . . . . 8 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
69		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑦On
70		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐵
71		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
72		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑦 +o
73	71, 72, 68	nfov 7388	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
74	70, 73	nfss 3926	. . . . . . . 8 ⊢ Ⅎ𝑦 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
75		oveq2 7366	. . . . . . . . 9 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
76	75	sseq2d 3966	. . . . . . . 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 3643	. . . . . . 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 496	. . . . . 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 17	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
80	64, 79	eqssd 3951	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = 𝐵)
81		oveq2 7366	. . . . . 6 ⊢ (𝑥 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑥) = (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
82	81	eqeq1d 2738	. . . . 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 584	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
85	84	ex 412	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
86		nnaword1 8557	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
87	86	adantlr 715	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
88		sseq2 3960	. . . 4 ⊢ ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ 𝐴 ⊆ 𝐵))
89	87, 88	syl5ibcom 245	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
90	89	rexlimdva 3137	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
91	85, 90	impbid 212	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060  {crab 3399   ⊆ wss 3901  ∅c0 4285  ∩ cint 4902  Ord word 6316  Oncon0 6317  suc csuc 6319  (class class class)co 7358  ωcom 7808   +_o coa 8394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-oadd 8401

This theorem is referenced by:  nnaordex  8566  eldifsucnn  8592  unfilem1  9205  ttrcltr  9625  hashdom  14302  precsexlem6  28208  precsexlem7  28209