Theorem oancom 9547

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

Step	Hyp	Ref	Expression
1		omex 9539	. . . 4 ⊢ ω ∈ V
2	1	sucid 6391	. . 3 ⊢ ω ∈ suc ω
3		omelon 9542	. . . 4 ⊢ ω ∈ On
4		1onn 8558	. . . 4 ⊢ 1o ∈ ω
5		oaabslem 8565	. . . 4 ⊢ ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω)
6	3, 4, 5	mp2an 692	. . 3 ⊢ (1o +o ω) = ω
7		oa1suc 8449	. . . 4 ⊢ (ω ∈ On → (ω +o 1o) = suc ω)
8	3, 7	ax-mp 5	. . 3 ⊢ (ω +o 1o) = suc ω
9	2, 6, 8	3eltr4i 2841	. 2 ⊢ (1o +o ω) ∈ (ω +o 1o)
10		1on 8400	. . . . 5 ⊢ 1o ∈ On
11		oacl 8453	. . . . 5 ⊢ ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On)
12	10, 3, 11	mp2an 692	. . . 4 ⊢ (1o +o ω) ∈ On
13		oacl 8453	. . . . 5 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On)
14	3, 10, 13	mp2an 692	. . . 4 ⊢ (ω +o 1o) ∈ On
15		onelpss 6347	. . . 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))))
16	12, 14, 15	mp2an 692	. . 3 ⊢ ((1o +o ω) ∈ (ω +o 1o) ↔ ((1o +o ω) ⊆ (ω +o 1o) ∧ (1o +o ω) ≠ (ω +o 1o)))
17	16	simprbi 496	. 2 ⊢ ((1o +o ω) ∈ (ω +o 1o) → (1o +o ω) ≠ (ω +o 1o))
18	9, 17	ax-mp 5	1 ⊢ (1o +o ω) ≠ (ω +o 1o)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ⊆ wss 3903  Oncon0 6307  suc csuc 6309  (class class class)co 7349  ωcom 7799  1_oc1o 8381   +_o coa 8385

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392

This theorem is referenced by:  oaomoencom  43294