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Theorem dfeldisj5 38703
Description: Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
Assertion
Ref Expression
dfeldisj5 ( ElDisj 𝐴 ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
Distinct variable group:   𝑢,𝐴,𝑣

Proof of Theorem dfeldisj5
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfeldisj4 38702 . 2 ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)
2 inecmo2 38338 . . . 4 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ∧ Rel E ) ↔ (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ∧ Rel E ))
3 relcnv 6125 . . . . 5 Rel E
43biantru 529 . . . 4 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ∧ Rel E ))
53biantru 529 . . . 4 (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ↔ (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ∧ Rel E ))
62, 4, 53bitr4i 303 . . 3 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ ∀𝑥∃*𝑢𝐴 𝑢 E 𝑥)
7 eccnvep 38264 . . . . . . . 8 (𝑢 ∈ V → [𝑢] E = 𝑢)
87elv 3483 . . . . . . 7 [𝑢] E = 𝑢
9 eccnvep 38264 . . . . . . . 8 (𝑣 ∈ V → [𝑣] E = 𝑣)
109elv 3483 . . . . . . 7 [𝑣] E = 𝑣
118, 10ineq12i 4226 . . . . . 6 ([𝑢] E ∩ [𝑣] E ) = (𝑢𝑣)
1211eqeq1i 2740 . . . . 5 (([𝑢] E ∩ [𝑣] E ) = ∅ ↔ (𝑢𝑣) = ∅)
1312orbi2i 912 . . . 4 ((𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
14132ralbii 3126 . . 3 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
15 brcnvep 38247 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
1615elv 3483 . . . . 5 (𝑢 E 𝑥𝑥𝑢)
1716rmobii 3386 . . . 4 (∃*𝑢𝐴 𝑢 E 𝑥 ↔ ∃*𝑢𝐴 𝑥𝑢)
1817albii 1816 . . 3 (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)
196, 14, 183bitr3i 301 . 2 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅) ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)
201, 19bitr4i 278 1 ( ElDisj 𝐴 ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wo 847  wal 1535   = wceq 1537  wral 3059  ∃*wrmo 3377  Vcvv 3478  cin 3962  c0 4339   class class class wbr 5148   E cep 5588  ccnv 5688  Rel wrel 5694  [cec 8742   ElDisj weldisj 38198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-coss 38393  df-cnvrefrel 38509  df-disjALTV 38687  df-eldisj 38689
This theorem is referenced by:  eqvreldisj2  38807
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