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Mirrors > Home > ILE Home > Th. List > el1o | 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 6388 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | eleq2i 2231 | . 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅}) |
3 | 0ex 4103 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | elsn2 3604 | . 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
5 | 2, 4 | bitri 183 | 1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∅c0 3404 {csn 3570 1oc1o 6368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-nul 4102 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-un 3115 df-nul 3405 df-sn 3576 df-suc 4343 df-1o 6375 |
This theorem is referenced by: 0lt1o 6399 map0e 6643 map1 6769 omp1eomlem 7050 ctmlemr 7064 ctssdclemn0 7066 exmidfodomrlemeldju 7146 exmidfodomrlemreseldju 7147 pw1on 7173 1tonninf 10365 |
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