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Theorem oancom 9675
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 9667 . . . 4 ω ∈ V
21sucid 6451 . . 3 ω ∈ suc ω
3 omelon 9670 . . . 4 ω ∈ On
4 1onn 8661 . . . 4 1o ∈ ω
5 oaabslem 8668 . . . 4 ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω)
63, 4, 5mp2an 691 . . 3 (1o +o ω) = ω
7 oa1suc 8552 . . . 4 (ω ∈ On → (ω +o 1o) = suc ω)
83, 7ax-mp 5 . . 3 (ω +o 1o) = suc ω
92, 6, 83eltr4i 2842 . 2 (1o +o ω) ∈ (ω +o 1o)
10 1on 8499 . . . . 5 1o ∈ On
11 oacl 8556 . . . . 5 ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On)
1210, 3, 11mp2an 691 . . . 4 (1o +o ω) ∈ On
13 oacl 8556 . . . . 5 ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On)
143, 10, 13mp2an 691 . . . 4 (ω +o 1o) ∈ On
15 onelpss 6409 . . . 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 691 . . 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 1534  wcel 2099  wne 2937  wss 3947  Oncon0 6369  suc csuc 6371  (class class class)co 7420  ωcom 7870  1oc1o 8480   +o coa 8484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740  ax-inf2 9665
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491
This theorem is referenced by:  oaomoencom  42746
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