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Mirrors > Home > ILE Home > Th. List > dif1o | GIF version |
Description: Two ways to say that 𝐴 is a nonzero number of the set 𝐵. (Contributed by Mario Carneiro, 21-May-2015.) |
Ref | Expression |
---|---|
dif1o | ⊢ (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 6013 | . . . 4 ⊢ 1𝑜 = {∅} | |
2 | 1 | difeq2i 3059 | . . 3 ⊢ (𝐵 ∖ 1𝑜) = (𝐵 ∖ {∅}) |
3 | 2 | eleq2i 2104 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ 𝐴 ∈ (𝐵 ∖ {∅})) |
4 | eldifsn 3495 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) | |
5 | 3, 4 | bitri 173 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∈ wcel 1393 ≠ wne 2204 ∖ cdif 2914 ∅c0 3224 {csn 3375 1𝑜c1o 5994 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-nul 3225 df-sn 3381 df-suc 4108 df-1o 6001 |
This theorem is referenced by: (None) |
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