Theorem omopthi 7624

Description: An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 12919. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Hypotheses

Ref	Expression
omopth.1	⊢ 𝐴 ∈ ω
omopth.2	⊢ 𝐵 ∈ ω
omopth.3	⊢ 𝐶 ∈ ω
omopth.4	⊢ 𝐷 ∈ ω

Assertion

Ref	Expression
omopthi	⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem omopthi

Step	Hyp	Ref	Expression
1		omopth.1	. . . . . . . . . . . . 13 ⊢ 𝐴 ∈ ω
2		omopth.2	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ ω
3	1, 2	nnacli 7581	. . . . . . . . . . . 12 ⊢ (𝐴 +𝑜 𝐵) ∈ ω
4	3	nnoni 6964	. . . . . . . . . . 11 ⊢ (𝐴 +𝑜 𝐵) ∈ On
5	4	onordi 5749	. . . . . . . . . 10 ⊢ Ord (𝐴 +𝑜 𝐵)
6		omopth.3	. . . . . . . . . . . . 13 ⊢ 𝐶 ∈ ω
7		omopth.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ ω
8	6, 7	nnacli 7581	. . . . . . . . . . . 12 ⊢ (𝐶 +𝑜 𝐷) ∈ ω
9	8	nnoni 6964	. . . . . . . . . . 11 ⊢ (𝐶 +𝑜 𝐷) ∈ On
10	9	onordi 5749	. . . . . . . . . 10 ⊢ Ord (𝐶 +𝑜 𝐷)
11		ordtri3 5676	. . . . . . . . . 10 ⊢ ((Ord (𝐴 +𝑜 𝐵) ∧ Ord (𝐶 +𝑜 𝐷)) → ((𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷) ↔ ¬ ((𝐴 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐷) ∨ (𝐶 +𝑜 𝐷) ∈ (𝐴 +𝑜 𝐵))))
12	5, 10, 11	mp2an 704	. . . . . . . . 9 ⊢ ((𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷) ↔ ¬ ((𝐴 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐷) ∨ (𝐶 +𝑜 𝐷) ∈ (𝐴 +𝑜 𝐵)))
13	12	con2bii 346	. . . . . . . 8 ⊢ (((𝐴 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐷) ∨ (𝐶 +𝑜 𝐷) ∈ (𝐴 +𝑜 𝐵)) ↔ ¬ (𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷))
14	1, 2, 8, 7	omopthlem2 7623	. . . . . . . . . 10 ⊢ ((𝐴 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐷) → ¬ (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵))
15		eqcom 2617	. . . . . . . . . 10 ⊢ ((((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ↔ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷))
16	14, 15	sylnib 317	. . . . . . . . 9 ⊢ ((𝐴 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐷) → ¬ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷))
17	6, 7, 3, 2	omopthlem2 7623	. . . . . . . . 9 ⊢ ((𝐶 +𝑜 𝐷) ∈ (𝐴 +𝑜 𝐵) → ¬ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷))
18	16, 17	jaoi 393	. . . . . . . 8 ⊢ (((𝐴 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐷) ∨ (𝐶 +𝑜 𝐷) ∈ (𝐴 +𝑜 𝐵)) → ¬ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷))
19	13, 18	sylbir 224	. . . . . . 7 ⊢ (¬ (𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷) → ¬ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷))
20	19	con4i 112	. . . . . 6 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → (𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷))
21		id 22	. . . . . . . . 9 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷))
22	20, 20	oveq12d 6567	. . . . . . . . . 10 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)))
23	22	oveq1d 6564	. . . . . . . . 9 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐷) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷))
24	21, 23	eqtr4d 2647	. . . . . . . 8 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐷))
25	3, 3	nnmcli 7582	. . . . . . . . 9 ⊢ ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω
26		nnacan 7595	. . . . . . . . 9 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐷 ∈ ω) → ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐷) ↔ 𝐵 = 𝐷))
27	25, 2, 7, 26	mp3an 1416	. . . . . . . 8 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐷) ↔ 𝐵 = 𝐷)
28	24, 27	sylib 207	. . . . . . 7 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → 𝐵 = 𝐷)
29	28	oveq2d 6565	. . . . . 6 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → (𝐶 +𝑜 𝐵) = (𝐶 +𝑜 𝐷))
30	20, 29	eqtr4d 2647	. . . . 5 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → (𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐵))
31		nnacom 7584	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 +𝑜 𝐴) = (𝐴 +𝑜 𝐵))
32	2, 1, 31	mp2an 704	. . . . 5 ⊢ (𝐵 +𝑜 𝐴) = (𝐴 +𝑜 𝐵)
33		nnacom 7584	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
34	2, 6, 33	mp2an 704	. . . . 5 ⊢ (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵)
35	30, 32, 34	3eqtr4g 2669	. . . 4 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → (𝐵 +𝑜 𝐴) = (𝐵 +𝑜 𝐶))
36		nnacan 7595	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 +𝑜 𝐴) = (𝐵 +𝑜 𝐶) ↔ 𝐴 = 𝐶))
37	2, 1, 6, 36	mp3an 1416	. . . 4 ⊢ ((𝐵 +𝑜 𝐴) = (𝐵 +𝑜 𝐶) ↔ 𝐴 = 𝐶)
38	35, 37	sylib 207	. . 3 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → 𝐴 = 𝐶)
39	38, 28	jca 553	. 2 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
40		oveq12 6558	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷))
41	40, 40	oveq12d 6567	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)))
42		simpr 476	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐵 = 𝐷)
43	41, 42	oveq12d 6567	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷))
44	39, 43	impbii 198	1 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ord word 5639  (class class class)co 6549  ωcom 6957   +_𝑜 coa 7444   ·_𝑜 comu 7445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452

This theorem is referenced by:  omopth  7625