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Mirrors > Home > MPE Home > Th. List > oancom | Structured version Visualization version GIF version |
Description: Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.) |
Ref | Expression |
---|---|
oancom | ⊢ (1𝑜 +𝑜 ω) ≠ (ω +𝑜 1𝑜) |
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) |
Colors of variables: wff setvar class |
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 1𝑜c1o 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) |
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