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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 8192 | . . . 4 ⊢ 1o = {∅} | |
2 | 1 | difeq2i 4020 | . . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅}) |
3 | 2 | eleq2i 2822 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅})) |
4 | eldifsn 4686 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) | |
5 | 3, 4 | bitri 278 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 ∅c0 4223 {csn 4527 1oc1o 8173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-sn 4528 df-suc 6197 df-1o 8180 |
This theorem is referenced by: ondif1 8206 brwitnlem 8212 oelim2 8301 oeeulem 8307 oeeui 8308 omabs 8354 cantnfp1lem3 9273 cantnfp1 9274 cantnflem1 9282 cantnflem3 9284 cantnflem4 9285 cnfcom3lem 9296 |
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