Theorem nnawordex 6124

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

Step	Hyp	Ref	Expression
1		nntri3or 6095	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
2	1	3adant3 958	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
3		nnaordex 6123	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
4		simpr 108	. . . . . . . 8 ⊢ ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (𝐴 +𝑜 𝑥) = 𝐵)
5	4	reximi 2458	. . . . . . 7 ⊢ (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
6	3, 5	syl6bi 161	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
7	6	3adant3 958	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
8		nna0 6076	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
9	8	3ad2ant1 959	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
10		eqeq2 2090	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((𝐴 +𝑜 ∅) = 𝐴 ↔ (𝐴 +𝑜 ∅) = 𝐵))
11	9, 10	syl5ibcom 153	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 = 𝐵 → (𝐴 +𝑜 ∅) = 𝐵))
12		peano1 4335	. . . . . . 7 ⊢ ∅ ∈ ω
13		oveq2 5540	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
14	13	eqeq1d 2089	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 ∅) = 𝐵))
15	14	rspcev 2701	. . . . . . 7 ⊢ ((∅ ∈ ω ∧ (𝐴 +𝑜 ∅) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
16	12, 15	mpan 414	. . . . . 6 ⊢ ((𝐴 +𝑜 ∅) = 𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
17	11, 16	syl6 33	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 = 𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
18		nntri1 6097	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))
19	18	biimp3a 1276	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → ¬ 𝐵 ∈ 𝐴)
20	19	pm2.21d 581	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ 𝐴 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
21	7, 17, 20	3jaod 1235	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
22	2, 21	mpd 13	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
23	22	3expia 1140	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
24		nnaword1 6109	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
25		sseq2 3021	. . . . 5 ⊢ ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ 𝐴 ⊆ 𝐵))
26	24, 25	syl5ibcom 153	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
27	26	rexlimdva 2477	. . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
28	27	adantr 270	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
29	23, 28	impbid 127	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ w3o 918   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∃wrex 2349   ⊆ wss 2973  ∅c0 3251  ωcom 4331  (class class class)co 5532   +_𝑜 coa 6021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028

This theorem is referenced by:  prarloclemn  6689