Theorem omopthi 8699

Description: An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 14309. (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 8652	. . . . . . . . . . . 12 ⊢ (𝐴 +o 𝐵) ∈ ω
4	3	nnoni 7894	. . . . . . . . . . 11 ⊢ (𝐴 +o 𝐵) ∈ On
5	4	onordi 6495	. . . . . . . . . 10 ⊢ Ord (𝐴 +o 𝐵)
6		omopth.3	. . . . . . . . . . . . 13 ⊢ 𝐶 ∈ ω
7		omopth.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ ω
8	6, 7	nnacli 8652	. . . . . . . . . . . 12 ⊢ (𝐶 +o 𝐷) ∈ ω
9	8	nnoni 7894	. . . . . . . . . . 11 ⊢ (𝐶 +o 𝐷) ∈ On
10	9	onordi 6495	. . . . . . . . . 10 ⊢ Ord (𝐶 +o 𝐷)
11		ordtri3 6420	. . . . . . . . . 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 692	. . . . . . . . 9 ⊢ ((𝐴 +o 𝐵) = (𝐶 +o 𝐷) ↔ ¬ ((𝐴 +o 𝐵) ∈ (𝐶 +o 𝐷) ∨ (𝐶 +o 𝐷) ∈ (𝐴 +o 𝐵)))
13	12	con2bii 357	. . . . . . . 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 8698	. . . . . . . . . 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 2744	. . . . . . . . . 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 328	. . . . . . . . 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 8698	. . . . . . . . 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 858	. . . . . . . 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 235	. . . . . . 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 114	. . . . . 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 7449	. . . . . . . . . 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 7446	. . . . . . . . 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 2780	. . . . . . . 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 8653	. . . . . . . . 9 ⊢ ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) ∈ ω
26		nnacan 8666	. . . . . . . . 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 1463	. . . . . . . 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 218	. . . . . . 7 ⊢ ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) → 𝐵 = 𝐷)
29	28	oveq2d 7447	. . . . . 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 2780	. . . . 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 8655	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 +o 𝐴) = (𝐴 +o 𝐵))
32	2, 1, 31	mp2an 692	. . . . 5 ⊢ (𝐵 +o 𝐴) = (𝐴 +o 𝐵)
33		nnacom 8655	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
34	2, 6, 33	mp2an 692	. . . . 5 ⊢ (𝐵 +o 𝐶) = (𝐶 +o 𝐵)
35	30, 32, 34	3eqtr4g 2802	. . . 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 8666	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 +o 𝐴) = (𝐵 +o 𝐶) ↔ 𝐴 = 𝐶))
37	2, 1, 6, 36	mp3an 1463	. . . 4 ⊢ ((𝐵 +o 𝐴) = (𝐵 +o 𝐶) ↔ 𝐴 = 𝐶)
38	35, 37	sylib 218	. . 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 511	. 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 7440	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 +o 𝐵) = (𝐶 +o 𝐷))
41	40, 40	oveq12d 7449	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) = ((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)))
42		simpr 484	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐵 = 𝐷)
43	41, 42	oveq12d 7449	. 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 209	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 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  Ord word 6383  (class class class)co 7431  ωcom 7887   +_o coa 8503   ·_o comu 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511

This theorem is referenced by:  omopth  8700