Theorem dfeldisj5 37586

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 37585	. 2 ⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
2		inecmo2 37220	. . . 4 ⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ∧ Rel ◡ E ) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ∧ Rel ◡ E ))
3		relcnv 6103	. . . . 5 ⊢ Rel ◡ E
4	3	biantru 530	. . . 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 530	. . . 4 ⊢ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ∧ Rel ◡ E ))
6	2, 4, 5	3bitr4i 302	. . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥)
7		eccnvep 37145	. . . . . . . 8 ⊢ (𝑢 ∈ V → [𝑢]◡ E = 𝑢)
8	7	elv 3480	. . . . . . 7 ⊢ [𝑢]◡ E = 𝑢
9		eccnvep 37145	. . . . . . . 8 ⊢ (𝑣 ∈ V → [𝑣]◡ E = 𝑣)
10	9	elv 3480	. . . . . . 7 ⊢ [𝑣]◡ E = 𝑣
11	8, 10	ineq12i 4210	. . . . . 6 ⊢ ([𝑢]◡ E ∩ [𝑣]◡ E ) = (𝑢 ∩ 𝑣)
12	11	eqeq1i 2737	. . . . 5 ⊢ (([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅ ↔ (𝑢 ∩ 𝑣) = ∅)
13	12	orbi2i 911	. . . 4 ⊢ ((𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))
14	13	2ralbii 3128	. . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))
15		brcnvep 37128	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢))
16	15	elv 3480	. . . . 5 ⊢ (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢)
17	16	rmobii 3384	. . . 4 ⊢ (∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ ∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
18	17	albii 1821	. . 3 ⊢ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
19	6, 14, 18	3bitr3i 300	. 2 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
20	1, 19	bitr4i 277	1 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∨ wo 845  ∀wal 1539   = wceq 1541  ∀wral 3061  ∃*wrmo 3375  Vcvv 3474   ∩ cin 3947  ∅c0 4322   class class class wbr 5148   E cep 5579  ^◡ccnv 5675  Rel wrel 5681  [cec 8700   ElDisj weldisj 37074

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8704  df-coss 37276  df-cnvrefrel 37392  df-disjALTV 37570  df-eldisj 37572

This theorem is referenced by:  eqvreldisj2  37690