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Theorem ral0 3350
 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 3256 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 585 . 2 (𝑥 ∈ ∅ → 𝜑)
32rgen 2391 1 𝑥 ∈ ∅ 𝜑
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1409  ∀wral 2323  ∅c0 3252 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-dif 2948  df-nul 3253 This theorem is referenced by:  0iin  3743  po0  4076  so0  4091  we0  4126  ord0  4156  mpt0  5054  ac6sfi  6383  rexfiuz  9816  bj-nntrans  10463
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