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Theorem eldisjeq 38105
Description: Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
Assertion
Ref Expression
eldisjeq (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))

Proof of Theorem eldisjeq
StepHypRef Expression
1 reseq2 5967 . . 3 (𝐴 = 𝐵 → ( E ↾ 𝐴) = ( E ↾ 𝐵))
21disjeqd 38100 . 2 (𝐴 = 𝐵 → ( Disj ( E ↾ 𝐴) ↔ Disj ( E ↾ 𝐵)))
3 df-eldisj 38071 . 2 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
4 df-eldisj 38071 . 2 ( ElDisj 𝐵 ↔ Disj ( E ↾ 𝐵))
52, 3, 43bitr4g 314 1 (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533   E cep 5570  ccnv 5666  cres 5669   Disj wdisjALTV 37571   ElDisj weldisj 37573
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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-coss 37775  df-cnvrefrel 37891  df-funALTV 38046  df-disjALTV 38069  df-eldisj 38071
This theorem is referenced by:  eldisjeqi  38106  eldisjeqd  38107  eqvrelqseqdisj2  38193
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