Theorem ral0 2358

Description: Vacuous universal quantification is always true.

Assertion

Ref	Expression
ral0	|- A.x e. (/) ph

Proof of Theorem ral0

Step	Hyp	Ref	Expression
1		noel 2284	. . 3 |- -. x e. (/)
2	1	pm2.21i 77	. 2 |- (x e. (/) -> ph)
3	2	rgen 1698	1 |- A.x e. (/) ph

Syntax hints:   e. wcel 958  A.wral 1645  (/)c0 2280

This theorem is referenced by:  0iin 2606  ixp0x 4359  xrsupsslem 6076  xrinfmsslem 6077  xrsup0 6097  0met 7825  chocnul 9292  emhgrat 10775

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-dif 2049  df-nul 2281

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