Theorem dfeldisj5a 38984

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 38983	. 2 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))
2		orcom 871	. . . 4 ⊢ ((𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑢 ∩ 𝑣) = ∅ ∨ 𝑢 = 𝑣))
3		neor 3023	. . . 4 ⊢ (((𝑢 ∩ 𝑣) = ∅ ∨ 𝑢 = 𝑣) ↔ ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
4	2, 3	bitri 275	. . 3 ⊢ ((𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
5	4	2ralbii 3110	. 2 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
6	1, 5	bitri 275	1 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   ≠ wne 2931  ∀wral 3050   ∩ cin 3899  ∅c0 4284   ElDisj weldisj 38391

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5518  df-eprel 5523  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8637  df-coss 38671  df-cnvrefrel 38777  df-disjALTV 38960  df-eldisj 38962

This theorem is referenced by:  eldisjim3  38985