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| Mirrors > Home > MPE Home > Th. List > dif20el | Structured version Visualization version GIF version | ||
| Description: An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.) |
| Ref | Expression |
|---|---|
| dif20el | ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ondif2 8540 | . . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴)) | |
| 2 | 1 | simprbi 496 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) → 1o ∈ 𝐴) |
| 3 | 0lt1o 8542 | . . 3 ⊢ ∅ ∈ 1o | |
| 4 | eldifi 4131 | . . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On) | |
| 5 | ontr1 6430 | . . . 4 ⊢ (𝐴 ∈ On → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴)) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴)) |
| 7 | 3, 6 | mpani 696 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) → (1o ∈ 𝐴 → ∅ ∈ 𝐴)) |
| 8 | 2, 7 | mpd 15 | 1 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∖ cdif 3948 ∅c0 4333 Oncon0 6384 1oc1o 8499 2oc2o 8500 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 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-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 df-1o 8506 df-2o 8507 |
| This theorem is referenced by: oeordi 8625 oeworde 8631 oelimcl 8638 oeeulem 8639 oeeui 8640 cantnfresb 43337 |
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