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Theorem oancom 9643
Description: Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.)
Assertion
Ref Expression
oancom (1o +o ω) ≠ (ω +o 1o)

Proof of Theorem oancom
StepHypRef Expression
1 omex 9635 . . . 4 ω ∈ V
21sucid 6437 . . 3 ω ∈ suc ω
3 omelon 9638 . . . 4 ω ∈ On
4 1onn 8636 . . . 4 1o ∈ ω
5 oaabslem 8643 . . . 4 ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω)
63, 4, 5mp2an 689 . . 3 (1o +o ω) = ω
7 oa1suc 8527 . . . 4 (ω ∈ On → (ω +o 1o) = suc ω)
83, 7ax-mp 5 . . 3 (ω +o 1o) = suc ω
92, 6, 83eltr4i 2838 . 2 (1o +o ω) ∈ (ω +o 1o)
10 1on 8474 . . . . 5 1o ∈ On
11 oacl 8531 . . . . 5 ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On)
1210, 3, 11mp2an 689 . . . 4 (1o +o ω) ∈ On
13 oacl 8531 . . . . 5 ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On)
143, 10, 13mp2an 689 . . . 4 (ω +o 1o) ∈ On
15 onelpss 6395 . . . 4 (((1o +o ω) ∈ On ∧ (ω +o 1o) ∈ On) → ((1o +o ω) ∈ (ω +o 1o) ↔ ((1o +o ω) ⊆ (ω +o 1o) ∧ (1o +o ω) ≠ (ω +o 1o))))
1612, 14, 15mp2an 689 . . 3 ((1o +o ω) ∈ (ω +o 1o) ↔ ((1o +o ω) ⊆ (ω +o 1o) ∧ (1o +o ω) ≠ (ω +o 1o)))
1716simprbi 496 . 2 ((1o +o ω) ∈ (ω +o 1o) → (1o +o ω) ≠ (ω +o 1o))
189, 17ax-mp 5 1 (1o +o ω) ≠ (ω +o 1o)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2932  wss 3941  Oncon0 6355  suc csuc 6357  (class class class)co 7402  ωcom 7849  1oc1o 8455   +o coa 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719  ax-inf2 9633
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466
This theorem is referenced by:  oaomoencom  42617
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