Theorem oancom 8431

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 8423	. . . 4 ⊢ ω ∈ V
2	1	sucid 5721	. . 3 ⊢ ω ∈ suc ω
3		omelon 8426	. . . 4 ⊢ ω ∈ On
4		1onn 7606	. . . 4 ⊢ 1𝑜 ∈ ω
5		oaabslem 7610	. . . 4 ⊢ ((ω ∈ On ∧ 1𝑜 ∈ ω) → (1𝑜 +𝑜 ω) = ω)
6	3, 4, 5	mp2an 704	. . 3 ⊢ (1𝑜 +𝑜 ω) = ω
7		oa1suc 7498	. . . 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 2701	. 2 ⊢ (1𝑜 +𝑜 ω) ∈ (ω +𝑜 1𝑜)
10		1on 7454	. . . . 5 ⊢ 1𝑜 ∈ On
11		oacl 7502	. . . . 5 ⊢ ((1𝑜 ∈ On ∧ ω ∈ On) → (1𝑜 +𝑜 ω) ∈ On)
12	10, 3, 11	mp2an 704	. . . 4 ⊢ (1𝑜 +𝑜 ω) ∈ On
13		oacl 7502	. . . . 5 ⊢ ((ω ∈ On ∧ 1𝑜 ∈ On) → (ω +𝑜 1𝑜) ∈ On)
14	3, 10, 13	mp2an 704	. . . 4 ⊢ (ω +𝑜 1𝑜) ∈ On
15		onelpss 5681	. . . 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 704	. . 3 ⊢ ((1𝑜 +𝑜 ω) ∈ (ω +𝑜 1𝑜) ↔ ((1𝑜 +𝑜 ω) ⊆ (ω +𝑜 1𝑜) ∧ (1𝑜 +𝑜 ω) ≠ (ω +𝑜 1𝑜)))
17	16	simprbi 479	. 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  Oncon0 5640  suc csuc 5642  (class class class)co 6549  ωcom 6957  1_𝑜c1o 7440   +_𝑜 coa 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451

This theorem is referenced by: (None)