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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 8416 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | eleq2i 2829 | . 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅}) |
3 | 0ex 5263 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | elsn2 4624 | . 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
5 | 2, 4 | bitri 274 | 1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∅c0 4281 {csn 4585 1oc1o 8402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5262 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3446 df-dif 3912 df-un 3914 df-nul 4282 df-sn 4586 df-suc 6322 df-1o 8409 |
This theorem is referenced by: ord1eln01 8439 ord2eln012 8440 0lt1o 8447 oelim2 8539 oeeulem 8545 oaabs2 8592 cantnff 9607 cnfcom3lem 9636 cfsuc 10190 pf1ind 21717 mavmul0 21897 cramer0 22035 cantnfresb 41634 omabs2 41641 omcl3g 41643 f1omo 46897 |
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