Theorem dfeldisj5a 39937

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 39936	. 2 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))
2		orcom 884	. . . 4 ⊢ ((𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑢 ∩ 𝑣) = ∅ ∨ 𝑢 = 𝑣))
3		neor 3051	. . . 4 ⊢ (((𝑢 ∩ 𝑣) = ∅ ∨ 𝑢 = 𝑣) ↔ ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
4	2, 3	bitri 278	. . 3 ⊢ ((𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
5	4	2ralbii 3141	. 2 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
6	1, 5	bitri 278	1 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570   ≠ wne 2959  ∀wral 3080   ∩ cin 4092  ∅c0 4281   ElDisj weldisj 39344

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 5252  ax-pr 5395

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 5171  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8744  df-coss 39624  df-cnvrefrel 39730  df-disjALTV 39913  df-eldisj 39915

This theorem is used by:  eldisjim3  39938