MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omopthi Structured version   Visualization version   GIF version

Theorem omopthi 8273
Description: An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 13618. (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 8229 . . . . . . . . . . . 12 (𝐴 +o 𝐵) ∈ ω
43nnoni 7576 . . . . . . . . . . 11 (𝐴 +o 𝐵) ∈ On
54onordi 6288 . . . . . . . . . 10 Ord (𝐴 +o 𝐵)
6 omopth.3 . . . . . . . . . . . . 13 𝐶 ∈ ω
7 omopth.4 . . . . . . . . . . . . 13 𝐷 ∈ ω
86, 7nnacli 8229 . . . . . . . . . . . 12 (𝐶 +o 𝐷) ∈ ω
98nnoni 7576 . . . . . . . . . . 11 (𝐶 +o 𝐷) ∈ On
109onordi 6288 . . . . . . . . . 10 Ord (𝐶 +o 𝐷)
11 ordtri3 6220 . . . . . . . . . 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 688 . . . . . . . . 9 ((𝐴 +o 𝐵) = (𝐶 +o 𝐷) ↔ ¬ ((𝐴 +o 𝐵) ∈ (𝐶 +o 𝐷) ∨ (𝐶 +o 𝐷) ∈ (𝐴 +o 𝐵)))
1312con2bii 359 . . . . . . . 8 (((𝐴 +o 𝐵) ∈ (𝐶 +o 𝐷) ∨ (𝐶 +o 𝐷) ∈ (𝐴 +o 𝐵)) ↔ ¬ (𝐴 +o 𝐵) = (𝐶 +o 𝐷))
141, 2, 8, 7omopthlem2 8272 . . . . . . . . . 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 2825 . . . . . . . . . 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 329 . . . . . . . . 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 8272 . . . . . . . . 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 851 . . . . . . . 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 236 . . . . . . 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 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))
2220, 20oveq12d 7163 . . . . . . . . . 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 7160 . . . . . . . . 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 2856 . . . . . . . 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 8230 . . . . . . . . 9 ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) ∈ ω
26 nnacan 8243 . . . . . . . . 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 1452 . . . . . . . 8 ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐷) ↔ 𝐵 = 𝐷)
2824, 27sylib 219 . . . . . . 7 ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) → 𝐵 = 𝐷)
2928oveq2d 7161 . . . . . 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 2856 . . . . 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 8232 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 +o 𝐴) = (𝐴 +o 𝐵))
322, 1, 31mp2an 688 . . . . 5 (𝐵 +o 𝐴) = (𝐴 +o 𝐵)
33 nnacom 8232 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
342, 6, 33mp2an 688 . . . . 5 (𝐵 +o 𝐶) = (𝐶 +o 𝐵)
3530, 32, 343eqtr4g 2878 . . . 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 8243 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 +o 𝐴) = (𝐵 +o 𝐶) ↔ 𝐴 = 𝐶))
372, 1, 6, 36mp3an 1452 . . . 4 ((𝐵 +o 𝐴) = (𝐵 +o 𝐶) ↔ 𝐴 = 𝐶)
3835, 37sylib 219 . . 3 ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) → 𝐴 = 𝐶)
3938, 28jca 512 . 2 ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
40 oveq12 7154 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 +o 𝐵) = (𝐶 +o 𝐷))
4140, 40oveq12d 7163 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) = ((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)))
42 simpr 485 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
4341, 42oveq12d 7163 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷))
4439, 43impbii 210 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 207  wa 396  wo 841   = wceq 1528  wcel 2105  Ord word 6183  (class class class)co 7145  ωcom 7569   +o coa 8088   ·o comu 8089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096
This theorem is referenced by:  omopth  8274
  Copyright terms: Public domain W3C validator