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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 8513 | . . 3 ⊢ 1o = {∅} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅}) | 
| 3 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 4 | 3 | elsn2 4665 | . 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) | 
| 5 | 2, 4 | bitri 275 | 1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∅c0 4333 {csn 4626 1oc1o 8499 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-suc 6390 df-1o 8506 | 
| This theorem is referenced by: ord1eln01 8534 ord2eln012 8535 0lt1o 8542 oelim2 8633 oeeulem 8639 oaabs2 8687 cantnff 9714 cnfcom3lem 9743 cfsuc 10297 pf1ind 22359 mavmul0 22558 cramer0 22696 cantnfresb 43337 omabs2 43345 omcl3g 43347 f1omo 48792 | 
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