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Theorem oancom 9595
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 9587 . . . 4 ω ∈ V
21sucid 6403 . . 3 ω ∈ suc ω
3 omelon 9590 . . . 4 ω ∈ On
4 1onn 8590 . . . 4 1o ∈ ω
5 oaabslem 8597 . . . 4 ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω)
63, 4, 5mp2an 691 . . 3 (1o +o ω) = ω
7 oa1suc 8481 . . . 4 (ω ∈ On → (ω +o 1o) = suc ω)
83, 7ax-mp 5 . . 3 (ω +o 1o) = suc ω
92, 6, 83eltr4i 2847 . 2 (1o +o ω) ∈ (ω +o 1o)
10 1on 8428 . . . . 5 1o ∈ On
11 oacl 8485 . . . . 5 ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On)
1210, 3, 11mp2an 691 . . . 4 (1o +o ω) ∈ On
13 oacl 8485 . . . . 5 ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On)
143, 10, 13mp2an 691 . . . 4 (ω +o 1o) ∈ On
15 onelpss 6361 . . . 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 498 . 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 397   = wceq 1542  wcel 2107  wne 2940  wss 3914  Oncon0 6321  suc csuc 6323  (class class class)co 7361  ωcom 7806  1oc1o 8409   +o coa 8413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676  ax-inf2 9585
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-oadd 8420
This theorem is referenced by:  oaomoencom  41699
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