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Theorem omopthi 7734
 Description: An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 13052. (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 = 𝐶𝐵 = 𝐷))

Proof of Theorem omopthi
StepHypRef Expression
1 omopth.1 . . . . . . . . . . . . 13 𝐴 ∈ ω
2 omopth.2 . . . . . . . . . . . . 13 𝐵 ∈ ω
31, 2nnacli 7691 . . . . . . . . . . . 12 (𝐴 +𝑜 𝐵) ∈ ω
43nnoni 7069 . . . . . . . . . . 11 (𝐴 +𝑜 𝐵) ∈ On
54onordi 5830 . . . . . . . . . 10 Ord (𝐴 +𝑜 𝐵)
6 omopth.3 . . . . . . . . . . . . 13 𝐶 ∈ ω
7 omopth.4 . . . . . . . . . . . . 13 𝐷 ∈ ω
86, 7nnacli 7691 . . . . . . . . . . . 12 (𝐶 +𝑜 𝐷) ∈ ω
98nnoni 7069 . . . . . . . . . . 11 (𝐶 +𝑜 𝐷) ∈ On
109onordi 5830 . . . . . . . . . 10 Ord (𝐶 +𝑜 𝐷)
11 ordtri3 5757 . . . . . . . . . 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))))
125, 10, 11mp2an 708 . . . . . . . . 9 ((𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷) ↔ ¬ ((𝐴 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐷) ∨ (𝐶 +𝑜 𝐷) ∈ (𝐴 +𝑜 𝐵)))
1312con2bii 347 . . . . . . . 8 (((𝐴 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐷) ∨ (𝐶 +𝑜 𝐷) ∈ (𝐴 +𝑜 𝐵)) ↔ ¬ (𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷))
141, 2, 8, 7omopthlem2 7733 . . . . . . . . . 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 2628 . . . . . . . . . 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))
1614, 15sylnib 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))
176, 7, 3, 2omopthlem2 7733 . . . . . . . . 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))
1816, 17jaoi 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))
1913, 18sylbir 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))
2019con4i 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))
2220, 20oveq12d 6665 . . . . . . . . . 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)))
2322oveq1d 6662 . . . . . . . . 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))
2421, 23eqtr4d 2658 . . . . . . . 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))
253, 3nnmcli 7692 . . . . . . . . 9 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω
26 nnacan 7705 . . . . . . . . 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))
2725, 2, 7, 26mp3an 1423 . . . . . . . 8 ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐷) ↔ 𝐵 = 𝐷)
2824, 27sylib 208 . . . . . . 7 ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → 𝐵 = 𝐷)
2928oveq2d 6663 . . . . . 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))
3020, 29eqtr4d 2658 . . . . 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 7694 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 +𝑜 𝐴) = (𝐴 +𝑜 𝐵))
322, 1, 31mp2an 708 . . . . 5 (𝐵 +𝑜 𝐴) = (𝐴 +𝑜 𝐵)
33 nnacom 7694 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
342, 6, 33mp2an 708 . . . . 5 (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵)
3530, 32, 343eqtr4g 2680 . . . 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 7705 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 +𝑜 𝐴) = (𝐵 +𝑜 𝐶) ↔ 𝐴 = 𝐶))
372, 1, 6, 36mp3an 1423 . . . 4 ((𝐵 +𝑜 𝐴) = (𝐵 +𝑜 𝐶) ↔ 𝐴 = 𝐶)
3835, 37sylib 208 . . 3 ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → 𝐴 = 𝐶)
3938, 28jca 554 . 2 ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
40 oveq12 6656 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 +𝑜 𝐵) = (𝐶 +𝑜 𝐷))
4140, 40oveq12d 6665 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)))
42 simpr 477 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
4341, 42oveq12d 6665 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷))
4439, 43impbii 199 1 ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1482   ∈ wcel 1989  Ord word 5720  (class class class)co 6647  ωcom 7062   +𝑜 coa 7554   ·𝑜 comu 7555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-omul 7562 This theorem is referenced by:  omopth  7735
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