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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 8513 | . . . 4 ⊢ 1o = {∅} | |
| 2 | 1 | difeq2i 4123 | . . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅}) | 
| 3 | 2 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅})) | 
| 4 | eldifsn 4786 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) | |
| 5 | 3, 4 | bitri 275 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ∅c0 4333 {csn 4626 1oc1o 8499 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-suc 6390 df-1o 8506 | 
| This theorem is referenced by: ondif1 8539 brwitnlem 8545 oelim2 8633 oeeulem 8639 oeeui 8640 omabs 8689 cantnfp1lem3 9720 cantnfp1 9721 cantnflem1 9729 cantnflem3 9731 cantnflem4 9732 cnfcom3lem 9743 | 
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