Theorem omopthi 7737

Description: An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 13057. (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 7694	. . . . . . . . . . . 12 ⊢ (𝐴 +𝑜 𝐵) ∈ ω
4	3	nnoni 7072	. . . . . . . . . . 11 ⊢ (𝐴 +𝑜 𝐵) ∈ On
5	4	onordi 5832	. . . . . . . . . 10 ⊢ Ord (𝐴 +𝑜 𝐵)
6		omopth.3	. . . . . . . . . . . . 13 ⊢ 𝐶 ∈ ω
7		omopth.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ ω
8	6, 7	nnacli 7694	. . . . . . . . . . . 12 ⊢ (𝐶 +𝑜 𝐷) ∈ ω
9	8	nnoni 7072	. . . . . . . . . . 11 ⊢ (𝐶 +𝑜 𝐷) ∈ On
10	9	onordi 5832	. . . . . . . . . 10 ⊢ Ord (𝐶 +𝑜 𝐷)
11		ordtri3 5759	. . . . . . . . . 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 708	. . . . . . . . 9 ⊢ ((𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷) ↔ ¬ ((𝐴 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐷) ∨ (𝐶 +𝑜 𝐷) ∈ (𝐴 +𝑜 𝐵)))
13	12	con2bii 347	. . . . . . . 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 7736	. . . . . . . . . 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 2629	. . . . . . . . . 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 318	. . . . . . . . 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 7736	. . . . . . . . 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 394	. . . . . . . 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 225	. . . . . . 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 113	. . . . . 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 6668	. . . . . . . . . 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 6665	. . . . . . . . 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 2659	. . . . . . . 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 7695	. . . . . . . . 9 ⊢ ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω
26		nnacan 7708	. . . . . . . . 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 1424	. . . . . . . 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 208	. . . . . . 7 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → 𝐵 = 𝐷)
29	28	oveq2d 6666	. . . . . 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 2659	. . . . 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 7697	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 +𝑜 𝐴) = (𝐴 +𝑜 𝐵))
32	2, 1, 31	mp2an 708	. . . . 5 ⊢ (𝐵 +𝑜 𝐴) = (𝐴 +𝑜 𝐵)
33		nnacom 7697	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
34	2, 6, 33	mp2an 708	. . . . 5 ⊢ (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵)
35	30, 32, 34	3eqtr4g 2681	. . . 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 7708	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 +𝑜 𝐴) = (𝐵 +𝑜 𝐶) ↔ 𝐴 = 𝐶))
37	2, 1, 6, 36	mp3an 1424	. . . 4 ⊢ ((𝐵 +𝑜 𝐴) = (𝐵 +𝑜 𝐶) ↔ 𝐴 = 𝐶)
38	35, 37	sylib 208	. . 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 554	. 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 6659	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷))
41	40, 40	oveq12d 6668	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)))
42		simpr 477	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐵 = 𝐷)
43	41, 42	oveq12d 6668	. 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 199	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 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Ord word 5722  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565

This theorem is referenced by:  omopth  7738