Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6326 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | eleq2i 2206 | . 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅}) |
3 | 0ex 4055 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | elsn2 3559 | . 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
5 | 2, 4 | bitri 183 | 1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∅c0 3363 {csn 3527 1oc1o 6306 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-nul 4054 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-un 3075 df-nul 3364 df-sn 3533 df-suc 4293 df-1o 6313 |
This theorem is referenced by: 0lt1o 6337 map0e 6580 map1 6706 omp1eomlem 6979 ctmlemr 6993 ctssdclemn0 6995 exmidfodomrlemeldju 7055 exmidfodomrlemreseldju 7056 1tonninf 10213 |
Copyright terms: Public domain | W3C validator |