Theorem omopthi 8286

Description: An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 13633. (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 8242	. . . . . . . . . . . 12 ⊢ (𝐴 +o 𝐵) ∈ ω
4	3	nnoni 7589	. . . . . . . . . . 11 ⊢ (𝐴 +o 𝐵) ∈ On
5	4	onordi 6297	. . . . . . . . . 10 ⊢ Ord (𝐴 +o 𝐵)
6		omopth.3	. . . . . . . . . . . . 13 ⊢ 𝐶 ∈ ω
7		omopth.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ ω
8	6, 7	nnacli 8242	. . . . . . . . . . . 12 ⊢ (𝐶 +o 𝐷) ∈ ω
9	8	nnoni 7589	. . . . . . . . . . 11 ⊢ (𝐶 +o 𝐷) ∈ On
10	9	onordi 6297	. . . . . . . . . 10 ⊢ Ord (𝐶 +o 𝐷)
11		ordtri3 6229	. . . . . . . . . 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 690	. . . . . . . . 9 ⊢ ((𝐴 +o 𝐵) = (𝐶 +o 𝐷) ↔ ¬ ((𝐴 +o 𝐵) ∈ (𝐶 +o 𝐷) ∨ (𝐶 +o 𝐷) ∈ (𝐴 +o 𝐵)))
13	12	con2bii 360	. . . . . . . 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 8285	. . . . . . . . . 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 2830	. . . . . . . . . 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 330	. . . . . . . . 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 8285	. . . . . . . . 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 853	. . . . . . . 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 237	. . . . . . 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 7176	. . . . . . . . . 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 7173	. . . . . . . . 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 2861	. . . . . . . 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 8243	. . . . . . . . 9 ⊢ ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) ∈ ω
26		nnacan 8256	. . . . . . . . 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 1457	. . . . . . . 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 220	. . . . . . 7 ⊢ ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) → 𝐵 = 𝐷)
29	28	oveq2d 7174	. . . . . 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 2861	. . . . 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 8245	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 +o 𝐴) = (𝐴 +o 𝐵))
32	2, 1, 31	mp2an 690	. . . . 5 ⊢ (𝐵 +o 𝐴) = (𝐴 +o 𝐵)
33		nnacom 8245	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
34	2, 6, 33	mp2an 690	. . . . 5 ⊢ (𝐵 +o 𝐶) = (𝐶 +o 𝐵)
35	30, 32, 34	3eqtr4g 2883	. . . 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 8256	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 +o 𝐴) = (𝐵 +o 𝐶) ↔ 𝐴 = 𝐶))
37	2, 1, 6, 36	mp3an 1457	. . . 4 ⊢ ((𝐵 +o 𝐴) = (𝐵 +o 𝐶) ↔ 𝐴 = 𝐶)
38	35, 37	sylib 220	. . 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 514	. 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 7167	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 +o 𝐵) = (𝐶 +o 𝐷))
41	40, 40	oveq12d 7176	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) = ((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)))
42		simpr 487	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐵 = 𝐷)
43	41, 42	oveq12d 7176	. 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 211	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 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  Ord word 6192  (class class class)co 7158  ωcom 7582   +_o coa 8101   ·_o comu 8102

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-omul 8109

This theorem is referenced by:  omopth  8287