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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 4250 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | 1 | a1d 25 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝐴 ≠ ∅)) |
3 | 2 | rexlimiv 3239 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ∅c0 4243 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rex 3112 df-dif 3884 df-nul 4244 |
This theorem is referenced by: 2reu4 4424 reusv2lem3 5266 eusvobj2 7128 isdrs2 17541 ismnd 17906 slwn0 18732 lbsexg 19929 iunconn 22033 grpon0 28285 filbcmb 35178 isbnd2 35221 rencldnfi 39762 iunconnlem2 41641 stoweidlem14 42656 hoidmvval0 43226 |
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