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Theorem dfeldisj5 36265
 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 36264 . 2 ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)
2 inecmo2 35921 . . . 4 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ∧ Rel E ) ↔ (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ∧ Rel E ))
3 relcnv 5938 . . . . 5 Rel E
43biantru 533 . . . 4 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ∧ Rel E ))
53biantru 533 . . . 4 (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ↔ (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ∧ Rel E ))
62, 4, 53bitr4i 306 . . 3 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢] E ∩ [𝑣] E ) = ∅) ↔ ∀𝑥∃*𝑢𝐴 𝑢 E 𝑥)
7 eccnvep 35850 . . . . . . . 8 (𝑢 ∈ V → [𝑢] E = 𝑢)
87elv 3447 . . . . . . 7 [𝑢] E = 𝑢
9 eccnvep 35850 . . . . . . . 8 (𝑣 ∈ V → [𝑣] E = 𝑣)
109elv 3447 . . . . . . 7 [𝑣] E = 𝑣
118, 10ineq12i 4140 . . . . . 6 ([𝑢] E ∩ [𝑣] E ) = (𝑢𝑣)
1211eqeq1i 2803 . . . . 5 (([𝑢] E ∩ [𝑣] E ) = ∅ ↔ (𝑢𝑣) = ∅)
1312orbi2i 910 . . . 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 35837 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
1615elv 3447 . . . . 5 (𝑢 E 𝑥𝑥𝑢)
1716rmobii 3350 . . . 4 (∃*𝑢𝐴 𝑢 E 𝑥 ↔ ∃*𝑢𝐴 𝑥𝑢)
1817albii 1821 . . 3 (∀𝑥∃*𝑢𝐴 𝑢 E 𝑥 ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)
196, 14, 183bitr3i 304 . 2 (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅) ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)
201, 19bitr4i 281 1 ( ElDisj 𝐴 ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538  ∀wral 3106  ∃*wrmo 3109  Vcvv 3442   ∩ cin 3882  ∅c0 4246   class class class wbr 5034   E cep 5433  ◡ccnv 5522  Rel wrel 5528  [cec 8288   ElDisj weldisj 35800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-id 5429  df-eprel 5434  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ec 8292  df-coss 35970  df-cnvrefrel 36076  df-disjALTV 36249  df-eldisj 36251 This theorem is referenced by: (None)
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