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Theorem ral0 3539
Description: Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
ral0 𝑥 ∈ ∅ 𝜑

Proof of Theorem ral0
StepHypRef Expression
1 noel 3441 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 647 . 2 (𝑥 ∈ ∅ → 𝜑)
32rgen 2543 1 𝑥 ∈ ∅ 𝜑
Colors of variables: wff set class
Syntax hints:  wcel 2160  wral 2468  c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-nul 3438
This theorem is referenced by:  0iin  3960  po0  4329  so0  4344  we0  4379  ord0  4409  omsinds  4639  mpt0  5362  iso0  5839  ixp0x  6752  ac6sfi  6926  fimax2gtri  6929  dcfi  7010  nnnninfeq2  7157  nninfisollem0  7158  finomni  7168  uzsinds  10473  seq3f1olemp  10533  rexfiuz  11030  fimaxre2  11267  2prm  12159  bj-nntrans  15161
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