Theorem omopthi 8571

Description: An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 14169. (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 8524	. . . . . . . . . . . 12 ⊢ (𝐴 +o 𝐵) ∈ ω
4	3	nnoni 7798	. . . . . . . . . . 11 ⊢ (𝐴 +o 𝐵) ∈ On
5	4	onordi 6415	. . . . . . . . . 10 ⊢ Ord (𝐴 +o 𝐵)
6		omopth.3	. . . . . . . . . . . . 13 ⊢ 𝐶 ∈ ω
7		omopth.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ ω
8	6, 7	nnacli 8524	. . . . . . . . . . . 12 ⊢ (𝐶 +o 𝐷) ∈ ω
9	8	nnoni 7798	. . . . . . . . . . 11 ⊢ (𝐶 +o 𝐷) ∈ On
10	9	onordi 6415	. . . . . . . . . 10 ⊢ Ord (𝐶 +o 𝐷)
11		ordtri3 6338	. . . . . . . . . 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 8570	. . . . . . . . . 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 2737	. . . . . . . . . 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 8570	. . . . . . . . 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 857	. . . . . . . 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 7359	. . . . . . . . . 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 7356	. . . . . . . . 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 2768	. . . . . . . 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 8525	. . . . . . . . 9 ⊢ ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) ∈ ω
26		nnacan 8538	. . . . . . . . 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 7357	. . . . . 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 2768	. . . . 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 8527	. . . . . 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 8527	. . . . . 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 2790	. . . 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 8538	. . . . 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 7350	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 +o 𝐵) = (𝐶 +o 𝐷))
41	40, 40	oveq12d 7359	. . 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 7359	. 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 847   = wceq 1541   ∈ wcel 2110  Ord word 6301  (class class class)co 7341  ωcom 7791   +_o coa 8377   ·_o comu 8378

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-omul 8385

This theorem is referenced by:  omopth  8572