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| Mirrors > Home > MPE Home > Th. List > rex0 | Structured version Visualization version GIF version | ||
| Description: Vacuous restricted existential quantification is false. (Contributed by NM, 15-Oct-2003.) |
| Ref | Expression |
|---|---|
| rex0 | ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | noel 4338 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
| 2 | 1 | pm2.21i 119 | . 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑) |
| 3 | 2 | nrex 3074 | 1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2108 ∃wrex 3070 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: reu0 4361 rmo0 4362 0iun 5063 0qs 8807 sup0riota 9505 cfeq0 10296 cfsuc 10297 hashge2el2difr 14520 cshws0 17139 addsrid 27997 muls01 28138 mulsrid 28139 elons2 28281 onaddscl 28286 onmulscl 28287 n0scut 28338 1p1e2s 28400 0ringirng 33739 dya2iocuni 34285 eulerpartlemgh 34380 pmapglb2xN 39774 elpadd0 39811 tfsconcatb0 43357 sprsymrelfvlem 47477 ipolub00 48882 |
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