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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 6429 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | eleq2i 2244 | . 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅}) |
3 | 0ex 4130 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | elsn2 3626 | . 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
5 | 2, 4 | bitri 184 | 1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∅c0 3422 {csn 3592 1oc1o 6409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-nul 4129 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-un 3133 df-nul 3423 df-sn 3598 df-suc 4371 df-1o 6416 |
This theorem is referenced by: 0lt1o 6440 map0e 6685 map1 6811 omp1eomlem 7092 ctmlemr 7106 ctssdclemn0 7108 exmidfodomrlemeldju 7197 exmidfodomrlemreseldju 7198 pw1on 7224 1tonninf 10437 |
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