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Theorem oancom 8586
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 8578 . . . 4 ω ∈ V
21sucid 5842 . . 3 ω ∈ suc ω
3 omelon 8581 . . . 4 ω ∈ On
4 1onn 7764 . . . 4 1𝑜 ∈ ω
5 oaabslem 7768 . . . 4 ((ω ∈ On ∧ 1𝑜 ∈ ω) → (1𝑜 +𝑜 ω) = ω)
63, 4, 5mp2an 708 . . 3 (1𝑜 +𝑜 ω) = ω
7 oa1suc 7656 . . . 4 (ω ∈ On → (ω +𝑜 1𝑜) = suc ω)
83, 7ax-mp 5 . . 3 (ω +𝑜 1𝑜) = suc ω
92, 6, 83eltr4i 2743 . 2 (1𝑜 +𝑜 ω) ∈ (ω +𝑜 1𝑜)
10 1on 7612 . . . . 5 1𝑜 ∈ On
11 oacl 7660 . . . . 5 ((1𝑜 ∈ On ∧ ω ∈ On) → (1𝑜 +𝑜 ω) ∈ On)
1210, 3, 11mp2an 708 . . . 4 (1𝑜 +𝑜 ω) ∈ On
13 oacl 7660 . . . . 5 ((ω ∈ On ∧ 1𝑜 ∈ On) → (ω +𝑜 1𝑜) ∈ On)
143, 10, 13mp2an 708 . . . 4 (ω +𝑜 1𝑜) ∈ On
15 onelpss 5802 . . . 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 708 . . 3 ((1𝑜 +𝑜 ω) ∈ (ω +𝑜 1𝑜) ↔ ((1𝑜 +𝑜 ω) ⊆ (ω +𝑜 1𝑜) ∧ (1𝑜 +𝑜 ω) ≠ (ω +𝑜 1𝑜)))
1716simprbi 479 . 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 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wss 3607  Oncon0 5761  suc csuc 5763  (class class class)co 6690  ωcom 7107  1𝑜c1o 7598   +𝑜 coa 7602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609
This theorem is referenced by: (None)
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