Theorem rex0 1717

Description: Existential quantification restricted to the empty set is false.

Assertion

Ref	Expression
rex0	⊢ ¬ ∃x ∈ ∅ φ

Proof of Theorem rex0

Step	Hyp	Ref	Expression
1		noel 1711	. . 3 ⊢ ¬ x ∈ ∅
2	1	pm2.21i 73	. 2 ⊢ (x ∈ ∅ → ¬ φ)
3	2	nrex 1270	1 ⊢ ¬ ∃x ∈ ∅ φ

Syntax hints:  ¬ wn 1   ∈ wcel 1092  ∃wrex 1202  ∅c0 1707

This theorem is referenced by:  cfsuc 3709

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-nul 1708

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