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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 8534 | . . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴)) | |
2 | 1 | simprbi 495 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) → 1o ∈ 𝐴) |
3 | 0lt1o 8536 | . . 3 ⊢ ∅ ∈ 1o | |
4 | eldifi 4126 | . . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On) | |
5 | ontr1 6424 | . . . 4 ⊢ (𝐴 ∈ On → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴)) |
7 | 3, 6 | mpani 694 | . 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 394 ∈ wcel 2099 ∖ cdif 3944 ∅c0 4325 Oncon0 6378 1oc1o 8491 2oc2o 8492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-tr 5273 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-ord 6381 df-on 6382 df-suc 6384 df-1o 8498 df-2o 8499 |
This theorem is referenced by: oeordi 8619 oeworde 8625 oelimcl 8632 oeeulem 8633 oeeui 8634 cantnfresb 43008 |
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