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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 9686 | . . . 4 ⊢ ω ∈ V | |
2 | 1 | sucid 6458 | . . 3 ⊢ ω ∈ suc ω |
3 | omelon 9689 | . . . 4 ⊢ ω ∈ On | |
4 | 1onn 8670 | . . . 4 ⊢ 1o ∈ ω | |
5 | oaabslem 8677 | . . . 4 ⊢ ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω) | |
6 | 3, 4, 5 | mp2an 690 | . . 3 ⊢ (1o +o ω) = ω |
7 | oa1suc 8561 | . . . 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 2839 | . 2 ⊢ (1o +o ω) ∈ (ω +o 1o) |
10 | 1on 8508 | . . . . 5 ⊢ 1o ∈ On | |
11 | oacl 8565 | . . . . 5 ⊢ ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On) | |
12 | 10, 3, 11 | mp2an 690 | . . . 4 ⊢ (1o +o ω) ∈ On |
13 | oacl 8565 | . . . . 5 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On) | |
14 | 3, 10, 13 | mp2an 690 | . . . 4 ⊢ (ω +o 1o) ∈ On |
15 | onelpss 6416 | . . . 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 690 | . . 3 ⊢ ((1o +o ω) ∈ (ω +o 1o) ↔ ((1o +o ω) ⊆ (ω +o 1o) ∧ (1o +o ω) ≠ (ω +o 1o))) |
17 | 16 | simprbi 495 | . 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 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ⊆ wss 3947 Oncon0 6376 suc csuc 6378 (class class class)co 7424 ωcom 7876 1oc1o 8489 +o coa 8493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 ax-inf2 9684 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-oadd 8500 |
This theorem is referenced by: oaomoencom 42983 |
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