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Theorem eldisjss 36613
Description: Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
Assertion
Ref Expression
eldisjss (𝐴𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))

Proof of Theorem eldisjss
StepHypRef Expression
1 ssres2 5894 . . 3 (𝐴𝐵 → ( E ↾ 𝐴) ⊆ ( E ↾ 𝐵))
21disjssd 36608 . 2 (𝐴𝐵 → ( Disj ( E ↾ 𝐵) → Disj ( E ↾ 𝐴)))
3 df-eldisj 36582 . 2 ( ElDisj 𝐵 ↔ Disj ( E ↾ 𝐵))
4 df-eldisj 36582 . 2 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
52, 3, 43imtr4g 299 1 (𝐴𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3881   E cep 5474  ccnv 5565  cres 5568   Disj wdisjALTV 36131   ElDisj weldisj 36133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-br 5069  df-opab 5131  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-coss 36301  df-cnvrefrel 36407  df-funALTV 36557  df-disjALTV 36580  df-eldisj 36582
This theorem is referenced by:  eldisjssi  36614  eldisjssd  36615
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