Theorem dfeldisj5a 39017

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 39016	. 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 3025	. . . 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 3112	. 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 2933  ∀wral 3052   ∩ cin 3901  ∅c0 4286   ElDisj weldisj 38424

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 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-coss 38704  df-cnvrefrel 38810  df-disjALTV 38993  df-eldisj 38995

This theorem is referenced by:  eldisjim3  39018