Theorem dfeldisj5 39948

Description: Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)

Assertion

Ref	Expression
dfeldisj5	⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))

Distinct variable group:   𝑢,𝐴,𝑣

Proof of Theorem dfeldisj5

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfeldisj4 39947	. 2 ⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
2		inecmo2 39491	. . . 4 ⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ∧ Rel ◡ E ) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ∧ Rel ◡ E ))
3		relcnv 6113	. . . . 5 ⊢ Rel ◡ E
4	3	biantru 539	. . . 4 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ∧ Rel ◡ E ))
5	3	biantru 539	. . . 4 ⊢ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ∧ Rel ◡ E ))
6	2, 4, 5	3bitr4i 306	. . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥)
7		eccnvep 39423	. . . . . . . 8 ⊢ (𝑢 ∈ V → [𝑢]◡ E = 𝑢)
8	7	elv 3459	. . . . . . 7 ⊢ [𝑢]◡ E = 𝑢
9		eccnvep 39423	. . . . . . . 8 ⊢ (𝑣 ∈ V → [𝑣]◡ E = 𝑣)
10	9	elv 3459	. . . . . . 7 ⊢ [𝑣]◡ E = 𝑣
11	8, 10	ineq12i 4118	. . . . . 6 ⊢ ([𝑢]◡ E ∩ [𝑣]◡ E ) = (𝑢 ∩ 𝑣)
12	11	eqeq1i 2768	. . . . 5 ⊢ (([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅ ↔ (𝑢 ∩ 𝑣) = ∅)
13	12	orbi2i 926	. . . 4 ⊢ ((𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))
14	13	2ralbii 3141	. . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))
15		brcnvep 39405	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢))
16	15	elv 3459	. . . . 5 ⊢ (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢)
17	16	rmobii 3377	. . . 4 ⊢ (∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ ∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
18	17	albii 1852	. . 3 ⊢ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
19	6, 14, 18	3bitr3i 304	. 2 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
20	1, 19	bitr4i 281	1 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570  ∀wral 3080  ∃*wrmo 3368  Vcvv 3454   ∩ cin 4092  ∅c0 4281   class class class wbr 5106   E cep 5555  ^◡ccnv 5655  Rel wrel 5661  [cec 8746   ElDisj weldisj 39356

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-eprel 5556  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ec 8750  df-coss 39636  df-cnvrefrel 39742  df-disjALTV 39925  df-eldisj 39927

This theorem is used by:  dfeldisj5a  39949  eqvreldisj2  40063