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Theorem oancom 8403
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 8395 . . . 4 ω ∈ V
21sucid 5702 . . 3 ω ∈ suc ω
3 omelon 8398 . . . 4 ω ∈ On
4 1onn 7578 . . . 4 1𝑜 ∈ ω
5 oaabslem 7582 . . . 4 ((ω ∈ On ∧ 1𝑜 ∈ ω) → (1𝑜 +𝑜 ω) = ω)
63, 4, 5mp2an 703 . . 3 (1𝑜 +𝑜 ω) = ω
7 oa1suc 7470 . . . 4 (ω ∈ On → (ω +𝑜 1𝑜) = suc ω)
83, 7ax-mp 5 . . 3 (ω +𝑜 1𝑜) = suc ω
92, 6, 83eltr4i 2695 . 2 (1𝑜 +𝑜 ω) ∈ (ω +𝑜 1𝑜)
10 1on 7426 . . . . 5 1𝑜 ∈ On
11 oacl 7474 . . . . 5 ((1𝑜 ∈ On ∧ ω ∈ On) → (1𝑜 +𝑜 ω) ∈ On)
1210, 3, 11mp2an 703 . . . 4 (1𝑜 +𝑜 ω) ∈ On
13 oacl 7474 . . . . 5 ((ω ∈ On ∧ 1𝑜 ∈ On) → (ω +𝑜 1𝑜) ∈ On)
143, 10, 13mp2an 703 . . . 4 (ω +𝑜 1𝑜) ∈ On
15 onelpss 5662 . . . 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 703 . . 3 ((1𝑜 +𝑜 ω) ∈ (ω +𝑜 1𝑜) ↔ ((1𝑜 +𝑜 ω) ⊆ (ω +𝑜 1𝑜) ∧ (1𝑜 +𝑜 ω) ≠ (ω +𝑜 1𝑜)))
1716simprbi 478 . 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 194  wa 382   = wceq 1474  wcel 1975  wne 2774  wss 3534  Oncon0 5621  suc csuc 5623  (class class class)co 6522  ωcom 6929  1𝑜c1o 7412   +𝑜 coa 7416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423
This theorem is referenced by: (None)
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