![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6455 | . . . 4 ⊢ 1o = {∅} | |
2 | 1 | difeq2i 3265 | . . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅}) |
3 | 2 | eleq2i 2256 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅})) |
4 | eldifsn 3734 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) | |
5 | 3, 4 | bitri 184 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2160 ≠ wne 2360 ∖ cdif 3141 ∅c0 3437 {csn 3607 1oc1o 6435 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-nul 3438 df-sn 3613 df-suc 4389 df-1o 6442 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |