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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 8837 | . . . 4 ⊢ ω ∈ V | |
2 | 1 | sucid 6055 | . . 3 ⊢ ω ∈ suc ω |
3 | omelon 8840 | . . . 4 ⊢ ω ∈ On | |
4 | 1onn 8003 | . . . 4 ⊢ 1o ∈ ω | |
5 | oaabslem 8007 | . . . 4 ⊢ ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω) | |
6 | 3, 4, 5 | mp2an 682 | . . 3 ⊢ (1o +o ω) = ω |
7 | oa1suc 7895 | . . . 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 2871 | . 2 ⊢ (1o +o ω) ∈ (ω +o 1o) |
10 | 1on 7850 | . . . . 5 ⊢ 1o ∈ On | |
11 | oacl 7899 | . . . . 5 ⊢ ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On) | |
12 | 10, 3, 11 | mp2an 682 | . . . 4 ⊢ (1o +o ω) ∈ On |
13 | oacl 7899 | . . . . 5 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On) | |
14 | 3, 10, 13 | mp2an 682 | . . . 4 ⊢ (ω +o 1o) ∈ On |
15 | onelpss 6016 | . . . 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 682 | . . 3 ⊢ ((1o +o ω) ∈ (ω +o 1o) ↔ ((1o +o ω) ⊆ (ω +o 1o) ∧ (1o +o ω) ≠ (ω +o 1o))) |
17 | 16 | simprbi 492 | . 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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ⊆ wss 3791 Oncon0 5976 suc csuc 5978 (class class class)co 6922 ωcom 7343 1oc1o 7836 +o coa 7840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 |
This theorem is referenced by: (None) |
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