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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 12475. (Contributed by AV, 17-Oct-2023.) | 
| Ref | Expression | 
|---|---|
| 1one2o | ⊢ 1o ≠ 2o | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 1onn 8679 | . . 3 ⊢ 1o ∈ ω | |
| 2 | omsucne 7907 | . . 3 ⊢ (1o ∈ ω → 1o ≠ suc 1o) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ 1o ≠ suc 1o | 
| 4 | df-2o 8508 | . 2 ⊢ 2o = suc 1o | |
| 5 | 3, 4 | neeqtrri 3013 | 1 ⊢ 1o ≠ 2o | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 ≠ wne 2939 suc csuc 6385 ωcom 7888 1oc1o 8500 2oc2o 8501 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-om 7889 df-1o 8507 df-2o 8508 | 
| This theorem is referenced by: gonanegoal 35358 satffunlem1lem1 35408 satffunlem2lem1 35410 ex-sategoelelomsuc 35432 ex-sategoelel12 35433 | 
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