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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 4247 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | pm2.21i 119 | . 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑) |
3 | 2 | nrex 3228 | 1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2111 ∃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-ral 3111 df-rex 3112 df-dif 3884 df-nul 4244 |
This theorem is referenced by: reu0 4272 rmo0 4273 0iun 4949 sup0riota 8913 cfeq0 9667 cfsuc 9668 hashge2el2difr 13835 cshws0 16427 dya2iocuni 31651 eulerpartlemgh 31746 0qs 35782 pmapglb2xN 37068 elpadd0 37105 sprsymrelfvlem 44007 |
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