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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 8127 | . . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴)) | |
2 | 1 | simprbi 499 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) → 1o ∈ 𝐴) |
3 | 0lt1o 8129 | . . 3 ⊢ ∅ ∈ 1o | |
4 | eldifi 4103 | . . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On) | |
5 | ontr1 6237 | . . . 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 398 ∈ wcel 2114 ∖ cdif 3933 ∅c0 4291 Oncon0 6191 1oc1o 8095 2oc2o 8096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 df-suc 6197 df-1o 8102 df-2o 8103 |
This theorem is referenced by: oeordi 8213 oeworde 8219 oelimcl 8226 oeeulem 8227 oeeui 8228 |
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