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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 7838 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | eleq2i 2897 | . 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅}) |
3 | 0ex 5013 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | elsn2 4431 | . 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
5 | 2, 4 | bitri 267 | 1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1658 ∈ wcel 2166 ∅c0 4143 {csn 4396 1oc1o 7818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-nul 5012 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-v 3415 df-dif 3800 df-un 3802 df-nul 4144 df-sn 4397 df-suc 5968 df-1o 7825 |
This theorem is referenced by: 0lt1o 7850 oelim2 7941 oeeulem 7947 oaabs2 7991 map0eOLD 8160 cantnff 8847 cnfcom3lem 8876 cfsuc 9393 pf1ind 20078 mavmul0 20725 cramer0 20865 |
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