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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 9683 | . . . 4 ⊢ ω ∈ V | |
| 2 | 1 | sucid 6466 | . . 3 ⊢ ω ∈ suc ω |
| 3 | omelon 9686 | . . . 4 ⊢ ω ∈ On | |
| 4 | 1onn 8678 | . . . 4 ⊢ 1o ∈ ω | |
| 5 | oaabslem 8685 | . . . 4 ⊢ ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω) | |
| 6 | 3, 4, 5 | mp2an 692 | . . 3 ⊢ (1o +o ω) = ω |
| 7 | oa1suc 8569 | . . . 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 2854 | . 2 ⊢ (1o +o ω) ∈ (ω +o 1o) |
| 10 | 1on 8518 | . . . . 5 ⊢ 1o ∈ On | |
| 11 | oacl 8573 | . . . . 5 ⊢ ((1o ∈ On ∧ ω ∈ On) → (1o +o ω) ∈ On) | |
| 12 | 10, 3, 11 | mp2an 692 | . . . 4 ⊢ (1o +o ω) ∈ On |
| 13 | oacl 8573 | . . . . 5 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω +o 1o) ∈ On) | |
| 14 | 3, 10, 13 | mp2an 692 | . . . 4 ⊢ (ω +o 1o) ∈ On |
| 15 | onelpss 6424 | . . . 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 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 Oncon0 6384 suc csuc 6386 (class class class)co 7431 ωcom 7887 1oc1o 8499 +o coa 8503 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 |
| This theorem is referenced by: oaomoencom 43330 |
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