MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oancom Structured version   Visualization version   GIF version

Theorem oancom 9521
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 9513 . . . 4 ω ∈ V
21sucid 6396 . . 3 ω ∈ suc ω
3 omelon 9516 . . . 4 ω ∈ On
4 1onn 8554 . . . 4 1o ∈ ω
5 oaabslem 8561 . . . 4 ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω)
63, 4, 5mp2an 691 . . 3 (1o +o ω) = ω
7 oa1suc 8445 . . . 4 (ω ∈ On → (ω +o 1o) = suc ω)
83, 7ax-mp 5 . . 3 (ω +o 1o) = suc ω
92, 6, 83eltr4i 2852 . 2 (1o +o ω) ∈ (ω +o 1o)
10 1on 8392 . . . . 5 1o ∈ On
11 oacl 8449 . . . . 5 ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On)
1210, 3, 11mp2an 691 . . . 4 (1o +o ω) ∈ On
13 oacl 8449 . . . . 5 ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On)
143, 10, 13mp2an 691 . . . 4 (ω +o 1o) ∈ On
15 onelpss 6354 . . . 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 2942  wss 3909  Oncon0 6314  suc csuc 6316  (class class class)co 7350  ωcom 7793  1oc1o 8373   +o coa 8377
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 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7663  ax-inf2 9511
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7353  df-oprab 7354  df-mpo 7355  df-om 7794  df-2nd 7913  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-1o 8380  df-oadd 8384
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator