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Theorem dfeldisj5 38193
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 38192 . 2 ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)
2 inecmo2 37828 . . . 4 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ∧ Rel E ) ↔ (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ∧ Rel E ))
3 relcnv 6108 . . . . 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 37754 . . . . . . . 8 (𝑢 ∈ V → [𝑢] E = 𝑢)
87elv 3477 . . . . . . 7 [𝑢] E = 𝑢
9 eccnvep 37754 . . . . . . . 8 (𝑣 ∈ V → [𝑣] E = 𝑣)
109elv 3477 . . . . . . 7 [𝑣] E = 𝑣
118, 10ineq12i 4210 . . . . . 6 ([𝑢] E ∩ [𝑣] E ) = (𝑢𝑣)
1211eqeq1i 2733 . . . . 5 (([𝑢] E ∩ [𝑣] E ) = ∅ ↔ (𝑢𝑣) = ∅)
1312orbi2i 911 . . . 4 ((𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
14132ralbii 3125 . . 3 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
15 brcnvep 37737 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
1615elv 3477 . . . . 5 (𝑢 E 𝑥𝑥𝑢)
1716rmobii 3381 . . . 4 (∃*𝑢𝐴 𝑢 E 𝑥 ↔ ∃*𝑢𝐴 𝑥𝑢)
1817albii 1814 . . 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 205  wa 395  wo 846  wal 1532   = wceq 1534  wral 3058  ∃*wrmo 3372  Vcvv 3471  cin 3946  c0 4323   class class class wbr 5148   E cep 5581  ccnv 5677  Rel wrel 5683  [cec 8723   ElDisj weldisj 37684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-eprel 5582  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-coss 37883  df-cnvrefrel 37999  df-disjALTV 38177  df-eldisj 38179
This theorem is referenced by:  eqvreldisj2  38297
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