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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 4121 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | 1 | a1d 25 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝐴 ≠ ∅)) |
3 | 2 | rexlimiv 3208 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 ≠ wne 2971 ∃wrex 3090 ∅c0 4115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-v 3387 df-dif 3772 df-nul 4116 |
This theorem is referenced by: reusv2lem3 5070 eusvobj2 6871 isdrs2 17254 ismnd 17612 slwn0 18343 lbsexg 19487 iunconn 21560 grpon0 27882 filbcmb 34023 isbnd2 34069 rencldnfi 38171 iunconnlem2 39931 stoweidlem14 40974 hoidmvval0 41547 2reu4 41967 |
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