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Theorem eldisjn0elb 38272
Description: Two forms of disjoint elements when the empty set is not an element of the class. (Contributed by Peter Mazsa, 31-Dec-2024.)
Assertion
Ref Expression
eldisjn0elb (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj ( E ↾ 𝐴) ∧ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))

Proof of Theorem eldisjn0elb
StepHypRef Expression
1 df-eldisj 38234 . 2 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
2 n0el3 38178 . 2 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
31, 2anbi12i 626 1 (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj ( E ↾ 𝐴) ∧ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394   = wceq 1533  wcel 2098  c0 4318   E cep 5575  ccnv 5671  dom cdm 5672  cres 5674   / cqs 8720   Disj wdisjALTV 37738   ElDisj weldisj 37740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8723  df-qs 8727  df-eldisj 38234
This theorem is referenced by:  mpet3  38363  cpet2  38364
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