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Theorem undefne0 7748
Description: The undefined value generated from a set is not empty. (Contributed by NM, 3-Sep-2018.)
Assertion
Ref Expression
undefne0 (𝑆𝑉 → (Undef‘𝑆) ≠ ∅)

Proof of Theorem undefne0
StepHypRef Expression
1 undefval 7745 . 2 (𝑆𝑉 → (Undef‘𝑆) = 𝒫 𝑆)
2 pwne0 5111 . . 3 𝒫 𝑆 ≠ ∅
32a1i 11 . 2 (𝑆𝑉 → 𝒫 𝑆 ≠ ∅)
41, 3eqnetrd 3034 1 (𝑆𝑉 → (Undef‘𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2050  wne 2967  c0 4178  𝒫 cpw 4422   cuni 4712  cfv 6188  Undefcund 7741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-undef 7742
This theorem is referenced by: (None)
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