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Mirrors > Home > MPE Home > Th. List > 1one2o | Structured version Visualization version GIF version |
Description: Ordinal one is not ordinal two. Analogous to 1ne2 12416. (Contributed by AV, 17-Oct-2023.) |
Ref | Expression |
---|---|
1one2o | ⊢ 1o ≠ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8635 | . . 3 ⊢ 1o ∈ ω | |
2 | omsucne 7870 | . . 3 ⊢ (1o ∈ ω → 1o ≠ suc 1o) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 1o ≠ suc 1o |
4 | df-2o 8463 | . 2 ⊢ 2o = suc 1o | |
5 | 3, 4 | neeqtrri 3014 | 1 ⊢ 1o ≠ 2o |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ≠ wne 2940 suc csuc 6363 ωcom 7851 1oc1o 8455 2oc2o 8456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-om 7852 df-1o 8462 df-2o 8463 |
This theorem is referenced by: gonanegoal 34331 satffunlem1lem1 34381 satffunlem2lem1 34383 ex-sategoelelomsuc 34405 ex-sategoelel12 34406 |
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