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Mirrors > Home > MPE Home > Th. List > el1o | Structured version Visualization version GIF version |
Description: Membership in ordinal one. (Contributed by NM, 5-Jan-2005.) |
Ref | Expression |
---|---|
el1o | ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 8211 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | eleq2i 2829 | . 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅}) |
3 | 0ex 5197 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | elsn2 4577 | . 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
5 | 2, 4 | bitri 278 | 1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∅c0 4234 {csn 4538 1oc1o 8192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-nul 5196 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3407 df-dif 3866 df-un 3868 df-nul 4235 df-sn 4539 df-suc 6216 df-1o 8199 |
This theorem is referenced by: 0lt1o 8228 oelim2 8320 oeeulem 8326 oaabs2 8371 cantnff 9286 cnfcom3lem 9315 cfsuc 9868 pf1ind 21268 mavmul0 21446 cramer0 21584 f1omo 45859 |
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