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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.) Avoid df-clel 2803, ax-8 2101. (Revised by GG, 2-Sep-2024.) |
Ref | Expression |
---|---|
rexn0 | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrex2 3063 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
2 | rzal 4503 | . . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
3 | 2 | con3i 154 | . . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ 𝐴 = ∅) |
4 | 1, 3 | sylbi 216 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ 𝐴 = ∅) |
5 | 4 | neqned 2937 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-ne 2931 df-ral 3052 df-rex 3061 df-dif 3949 df-nul 4323 |
This theorem is referenced by: 2reu4 4521 reusv2lem3 5396 eusvobj2 7408 isdrs2 18326 ismnd 18725 slwn0 19609 lbsexg 21141 iunconn 23420 sltn0 27925 grpon0 30432 filbcmb 37454 isbnd2 37497 rencldnfi 42515 iunconnlem2 44648 stoweidlem14 45671 hoidmvval0 46244 thinciso 48417 |
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