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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 6232 | . . . 4 ⊢ 1o = {∅} | |
2 | 1 | difeq2i 3130 | . . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅}) |
3 | 2 | eleq2i 2161 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅})) |
4 | eldifsn 3589 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) | |
5 | 3, 4 | bitri 183 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1445 ≠ wne 2262 ∖ cdif 3010 ∅c0 3302 {csn 3466 1oc1o 6212 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rab 2379 df-v 2635 df-dif 3015 df-un 3017 df-nul 3303 df-sn 3472 df-suc 4222 df-1o 6219 |
This theorem is referenced by: (None) |
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