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Theorem dfeldisj5 38723
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 38722 . 2 ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)
2 inecmo2 38358 . . . 4 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ∧ Rel E ) ↔ (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ∧ Rel E ))
3 relcnv 6121 . . . . 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 38284 . . . . . . . 8 (𝑢 ∈ V → [𝑢] E = 𝑢)
87elv 3484 . . . . . . 7 [𝑢] E = 𝑢
9 eccnvep 38284 . . . . . . . 8 (𝑣 ∈ V → [𝑣] E = 𝑣)
109elv 3484 . . . . . . 7 [𝑣] E = 𝑣
118, 10ineq12i 4217 . . . . . 6 ([𝑢] E ∩ [𝑣] E ) = (𝑢𝑣)
1211eqeq1i 2741 . . . . 5 (([𝑢] E ∩ [𝑣] E ) = ∅ ↔ (𝑢𝑣) = ∅)
1312orbi2i 912 . . . 4 ((𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
14132ralbii 3127 . . 3 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
15 brcnvep 38267 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
1615elv 3484 . . . . 5 (𝑢 E 𝑥𝑥𝑢)
1716rmobii 3387 . . . 4 (∃*𝑢𝐴 𝑢 E 𝑥 ↔ ∃*𝑢𝐴 𝑥𝑢)
1817albii 1818 . . 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 1537   = wceq 1539  wral 3060  ∃*wrmo 3378  Vcvv 3479  cin 3949  c0 4332   class class class wbr 5142   E cep 5582  ccnv 5683  Rel wrel 5689  [cec 8744   ElDisj weldisj 38219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-eprel 5583  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748  df-coss 38413  df-cnvrefrel 38529  df-disjALTV 38707  df-eldisj 38709
This theorem is referenced by:  eqvreldisj2  38827
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