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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 9587 | . . . 4 ⊢ ω ∈ V | |
2 | 1 | sucid 6403 | . . 3 ⊢ ω ∈ suc ω |
3 | omelon 9590 | . . . 4 ⊢ ω ∈ On | |
4 | 1onn 8590 | . . . 4 ⊢ 1o ∈ ω | |
5 | oaabslem 8597 | . . . 4 ⊢ ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω) | |
6 | 3, 4, 5 | mp2an 691 | . . 3 ⊢ (1o +o ω) = ω |
7 | oa1suc 8481 | . . . 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 2847 | . 2 ⊢ (1o +o ω) ∈ (ω +o 1o) |
10 | 1on 8428 | . . . . 5 ⊢ 1o ∈ On | |
11 | oacl 8485 | . . . . 5 ⊢ ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On) | |
12 | 10, 3, 11 | mp2an 691 | . . . 4 ⊢ (1o +o ω) ∈ On |
13 | oacl 8485 | . . . . 5 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On) | |
14 | 3, 10, 13 | mp2an 691 | . . . 4 ⊢ (ω +o 1o) ∈ On |
15 | onelpss 6361 | . . . 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 691 | . . 3 ⊢ ((1o +o ω) ∈ (ω +o 1o) ↔ ((1o +o ω) ⊆ (ω +o 1o) ∧ (1o +o ω) ≠ (ω +o 1o))) |
17 | 16 | simprbi 498 | . 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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ⊆ wss 3914 Oncon0 6321 suc csuc 6323 (class class class)co 7361 ωcom 7806 1oc1o 8409 +o coa 8413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 ax-inf2 9585 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 |
This theorem is referenced by: oaomoencom 41699 |
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