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Mirrors > Home > MPE Home > Th. List > rexn0 | Structured version Visualization version GIF version |
Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
Ref | Expression |
---|---|
rexn0 | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4302 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | 1 | a1d 25 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝐴 ≠ ∅)) |
3 | 2 | rexlimiv 3282 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 ∅c0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ne 3019 df-ral 3145 df-rex 3146 df-dif 3941 df-nul 4294 |
This theorem is referenced by: 2reu4 4468 reusv2lem3 5303 eusvobj2 7151 isdrs2 17551 ismnd 17916 slwn0 18742 lbsexg 19938 iunconn 22038 grpon0 28281 filbcmb 35017 isbnd2 35063 rencldnfi 39425 iunconnlem2 41276 stoweidlem14 42306 hoidmvval0 42876 |
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