Theorem omopthi 8588

Description: An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 14221. (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 8541	. . . . . . . . . . . 12 ⊢ (𝐴 +o 𝐵) ∈ ω
4	3	nnoni 7815	. . . . . . . . . . 11 ⊢ (𝐴 +o 𝐵) ∈ On
5	4	onordi 6428	. . . . . . . . . 10 ⊢ Ord (𝐴 +o 𝐵)
6		omopth.3	. . . . . . . . . . . . 13 ⊢ 𝐶 ∈ ω
7		omopth.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ ω
8	6, 7	nnacli 8541	. . . . . . . . . . . 12 ⊢ (𝐶 +o 𝐷) ∈ ω
9	8	nnoni 7815	. . . . . . . . . . 11 ⊢ (𝐶 +o 𝐷) ∈ On
10	9	onordi 6428	. . . . . . . . . 10 ⊢ Ord (𝐶 +o 𝐷)
11		ordtri3 6351	. . . . . . . . . 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 693	. . . . . . . . 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 8587	. . . . . . . . . 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 8587	. . . . . . . . 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 7376	. . . . . . . . . 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 7373	. . . . . . . . 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 2775	. . . . . . . 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 8542	. . . . . . . . 9 ⊢ ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) ∈ ω
26		nnacan 8555	. . . . . . . . 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 1464	. . . . . . . 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 7374	. . . . . 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 2775	. . . . 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 8544	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 +o 𝐴) = (𝐴 +o 𝐵))
32	2, 1, 31	mp2an 693	. . . . 5 ⊢ (𝐵 +o 𝐴) = (𝐴 +o 𝐵)
33		nnacom 8544	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
34	2, 6, 33	mp2an 693	. . . . 5 ⊢ (𝐵 +o 𝐶) = (𝐶 +o 𝐵)
35	30, 32, 34	3eqtr4g 2797	. . . 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 8555	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 +o 𝐴) = (𝐵 +o 𝐶) ↔ 𝐴 = 𝐶))
37	2, 1, 6, 36	mp3an 1464	. . . 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 7367	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 +o 𝐵) = (𝐶 +o 𝐷))
41	40, 40	oveq12d 7376	. . 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 7376	. 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 1542   ∈ wcel 2114  Ord word 6314  (class class class)co 7358  ωcom 7808   +_o coa 8393   ·_o comu 8394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-oadd 8400  df-omul 8401

This theorem is referenced by:  omopth  8589