Theorem dfeldisj5a 39196

Description: Alternate definition of the disjoint elementhood predicate. Members of 𝐴 are pairwise disjoint: if two members overlap, they are equal. (Contributed by Peter Mazsa, 19-Sep-2021.)

Assertion

Ref	Expression
dfeldisj5a	⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))

Distinct variable group:   𝑢,𝐴,𝑣

Proof of Theorem dfeldisj5a

Step	Hyp	Ref	Expression
1		dfeldisj5 39195	. 2 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))
2		orcom 877	. . . 4 ⊢ ((𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑢 ∩ 𝑣) = ∅ ∨ 𝑢 = 𝑣))
3		neor 3028	. . . 4 ⊢ (((𝑢 ∩ 𝑣) = ∅ ∨ 𝑢 = 𝑣) ↔ ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
4	2, 3	bitri 277	. . 3 ⊢ ((𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
5	4	2ralbii 3116	. 2 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
6	1, 5	bitri 277	1 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 854   = wceq 1548   ≠ wne 2936  ∀wral 3055   ∩ cin 3884  ∅c0 4264   ElDisj weldisj 38603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8639  df-coss 38883  df-cnvrefrel 38989  df-disjALTV 39172  df-eldisj 39174

This theorem is referenced by:  eldisjim3  39197