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Theorem oancom 9691
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 9683 . . . 4 ω ∈ V
21sucid 6466 . . 3 ω ∈ suc ω
3 omelon 9686 . . . 4 ω ∈ On
4 1onn 8678 . . . 4 1o ∈ ω
5 oaabslem 8685 . . . 4 ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω)
63, 4, 5mp2an 692 . . 3 (1o +o ω) = ω
7 oa1suc 8569 . . . 4 (ω ∈ On → (ω +o 1o) = suc ω)
83, 7ax-mp 5 . . 3 (ω +o 1o) = suc ω
92, 6, 83eltr4i 2854 . 2 (1o +o ω) ∈ (ω +o 1o)
10 1on 8518 . . . . 5 1o ∈ On
11 oacl 8573 . . . . 5 ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On)
1210, 3, 11mp2an 692 . . . 4 (1o +o ω) ∈ On
13 oacl 8573 . . . . 5 ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On)
143, 10, 13mp2an 692 . . . 4 (ω +o 1o) ∈ On
15 onelpss 6424 . . . 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 692 . . 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 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wss 3951  Oncon0 6384  suc csuc 6386  (class class class)co 7431  ωcom 7887  1oc1o 8499   +o coa 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510
This theorem is referenced by:  oaomoencom  43330
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