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Theorem dfeldisj5 38677
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 38676 . 2 ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)
2 inecmo2 38312 . . . 4 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ∧ Rel E ) ↔ (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ∧ Rel E ))
3 relcnv 6134 . . . . 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 38238 . . . . . . . 8 (𝑢 ∈ V → [𝑢] E = 𝑢)
87elv 3493 . . . . . . 7 [𝑢] E = 𝑢
9 eccnvep 38238 . . . . . . . 8 (𝑣 ∈ V → [𝑣] E = 𝑣)
109elv 3493 . . . . . . 7 [𝑣] E = 𝑣
118, 10ineq12i 4239 . . . . . 6 ([𝑢] E ∩ [𝑣] E ) = (𝑢𝑣)
1211eqeq1i 2745 . . . . 5 (([𝑢] E ∩ [𝑣] E ) = ∅ ↔ (𝑢𝑣) = ∅)
1312orbi2i 911 . . . 4 ((𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
14132ralbii 3134 . . 3 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
15 brcnvep 38221 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
1615elv 3493 . . . . 5 (𝑢 E 𝑥𝑥𝑢)
1716rmobii 3396 . . . 4 (∃*𝑢𝐴 𝑢 E 𝑥 ↔ ∃*𝑢𝐴 𝑥𝑢)
1817albii 1817 . . 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 846  wal 1535   = wceq 1537  wral 3067  ∃*wrmo 3387  Vcvv 3488  cin 3975  c0 4352   class class class wbr 5166   E cep 5598  ccnv 5699  Rel wrel 5705  [cec 8761   ElDisj weldisj 38171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765  df-coss 38367  df-cnvrefrel 38483  df-disjALTV 38661  df-eldisj 38663
This theorem is referenced by:  eqvreldisj2  38781
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