![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8538 | . . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴)) | |
2 | 1 | simprbi 496 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) → 1o ∈ 𝐴) |
3 | 0lt1o 8540 | . . 3 ⊢ ∅ ∈ 1o | |
4 | eldifi 4140 | . . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On) | |
5 | ontr1 6431 | . . . 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 2105 ∖ cdif 3959 ∅c0 4338 Oncon0 6385 1oc1o 8497 2oc2o 8498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 df-suc 6391 df-1o 8504 df-2o 8505 |
This theorem is referenced by: oeordi 8623 oeworde 8629 oelimcl 8636 oeeulem 8637 oeeui 8638 cantnfresb 43313 |
Copyright terms: Public domain | W3C validator |