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Mirrors > Home > MPE Home > Th. List > dif1o | Structured version Visualization version 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 8419 | . . . 4 ⊢ 1o = {∅} | |
2 | 1 | difeq2i 4079 | . . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅}) |
3 | 2 | eleq2i 2829 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅})) |
4 | eldifsn 4747 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) | |
5 | 3, 4 | bitri 274 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3907 ∅c0 4282 {csn 4586 1oc1o 8405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-nul 4283 df-sn 4587 df-suc 6323 df-1o 8412 |
This theorem is referenced by: ondif1 8447 brwitnlem 8453 oelim2 8542 oeeulem 8548 oeeui 8549 omabs 8597 cantnfp1lem3 9616 cantnfp1 9617 cantnflem1 9625 cantnflem3 9627 cantnflem4 9628 cnfcom3lem 9639 |
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