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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 7866 | . . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴)) | |
2 | 1 | simprbi 492 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) → 1o ∈ 𝐴) |
3 | 0lt1o 7868 | . . 3 ⊢ ∅ ∈ 1o | |
4 | eldifi 3954 | . . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On) | |
5 | ontr1 6022 | . . . 4 ⊢ (𝐴 ∈ On → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴)) |
7 | 3, 6 | mpani 686 | . 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 386 ∈ wcel 2106 ∖ cdif 3788 ∅c0 4140 Oncon0 5976 1oc1o 7836 2oc2o 7837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-tr 4988 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-ord 5979 df-on 5980 df-suc 5982 df-1o 7843 df-2o 7844 |
This theorem is referenced by: oeordi 7951 oeworde 7957 oelimcl 7964 oeeulem 7965 oeeui 7966 |
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