Theorem oancom 8548

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 8540	. . . 4 ⊢ ω ∈ V
2	1	sucid 5804	. . 3 ⊢ ω ∈ suc ω
3		omelon 8543	. . . 4 ⊢ ω ∈ On
4		1onn 7719	. . . 4 ⊢ 1𝑜 ∈ ω
5		oaabslem 7723	. . . 4 ⊢ ((ω ∈ On ∧ 1𝑜 ∈ ω) → (1𝑜 +𝑜 ω) = ω)
6	3, 4, 5	mp2an 708	. . 3 ⊢ (1𝑜 +𝑜 ω) = ω
7		oa1suc 7611	. . . 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 2714	. 2 ⊢ (1𝑜 +𝑜 ω) ∈ (ω +𝑜 1𝑜)
10		1on 7567	. . . . 5 ⊢ 1𝑜 ∈ On
11		oacl 7615	. . . . 5 ⊢ ((1𝑜 ∈ On ∧ ω ∈ On) → (1𝑜 +𝑜 ω) ∈ On)
12	10, 3, 11	mp2an 708	. . . 4 ⊢ (1𝑜 +𝑜 ω) ∈ On
13		oacl 7615	. . . . 5 ⊢ ((ω ∈ On ∧ 1𝑜 ∈ On) → (ω +𝑜 1𝑜) ∈ On)
14	3, 10, 13	mp2an 708	. . . 4 ⊢ (ω +𝑜 1𝑜) ∈ On
15		onelpss 5764	. . . 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 708	. . 3 ⊢ ((1𝑜 +𝑜 ω) ∈ (ω +𝑜 1𝑜) ↔ ((1𝑜 +𝑜 ω) ⊆ (ω +𝑜 1𝑜) ∧ (1𝑜 +𝑜 ω) ≠ (ω +𝑜 1𝑜)))
17	16	simprbi 480	. 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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ⊆ wss 3574  Oncon0 5723  suc csuc 5725  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553   +_𝑜 coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564

This theorem is referenced by: (None)