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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 9681 | . . . 4 ⊢ ω ∈ V | |
2 | 1 | sucid 6468 | . . 3 ⊢ ω ∈ suc ω |
3 | omelon 9684 | . . . 4 ⊢ ω ∈ On | |
4 | 1onn 8677 | . . . 4 ⊢ 1o ∈ ω | |
5 | oaabslem 8684 | . . . 4 ⊢ ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω) | |
6 | 3, 4, 5 | mp2an 692 | . . 3 ⊢ (1o +o ω) = ω |
7 | oa1suc 8568 | . . . 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 2852 | . 2 ⊢ (1o +o ω) ∈ (ω +o 1o) |
10 | 1on 8517 | . . . . 5 ⊢ 1o ∈ On | |
11 | oacl 8572 | . . . . 5 ⊢ ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On) | |
12 | 10, 3, 11 | mp2an 692 | . . . 4 ⊢ (1o +o ω) ∈ On |
13 | oacl 8572 | . . . . 5 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On) | |
14 | 3, 10, 13 | mp2an 692 | . . . 4 ⊢ (ω +o 1o) ∈ On |
15 | onelpss 6426 | . . . 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 692 | . . 3 ⊢ ((1o +o ω) ∈ (ω +o 1o) ↔ ((1o +o ω) ⊆ (ω +o 1o) ∧ (1o +o ω) ≠ (ω +o 1o))) |
17 | 16 | simprbi 496 | . 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 Oncon0 6386 suc csuc 6388 (class class class)co 7431 ωcom 7887 1oc1o 8498 +o coa 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 |
This theorem is referenced by: oaomoencom 43307 |
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