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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 8105 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | eleq2i 2901 | . 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅}) |
3 | 0ex 5202 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | elsn2 4594 | . 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
5 | 2, 4 | bitri 276 | 1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∅c0 4288 {csn 4557 1oc1o 8084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-un 3938 df-nul 4289 df-sn 4558 df-suc 6190 df-1o 8091 |
This theorem is referenced by: 0lt1o 8118 oelim2 8210 oeeulem 8216 oaabs2 8261 cantnff 9125 cnfcom3lem 9154 cfsuc 9667 pf1ind 20446 mavmul0 21089 cramer0 21227 |
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