Theorem dfdisjALTV5a 39144

Description: Alternate definition of the disjoint relation predicate. Disj 𝑅 means: different domain generators have disjoint cosets (unless the generators are equal), plus Rel 𝑅 for relation-typedness. This is the characterization that makes canonicity/uniqueness arguments modular. It is the starting point for the entire "Disj ↔ unique representative per block" pipeline that feeds into Disjs, see dfdisjs7 39284. (Contributed by Peter Mazsa, 3-Feb-2026.)

Assertion

Ref	Expression
dfdisjALTV5a	⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) ∧ Rel 𝑅))

Distinct variable group:   𝑢,𝑅,𝑣

Proof of Theorem dfdisjALTV5a

Step	Hyp	Ref	Expression
1		dfdisjALTV5 39143	. 2 ⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))
2		orcom 871	. . . 4 ⊢ ((𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅ ∨ 𝑢 = 𝑣))
3		neor 3025	. . . 4 ⊢ ((([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅ ∨ 𝑢 = 𝑣) ↔ (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣))
4	2, 3	bitri 275	. . 3 ⊢ ((𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣))
5	4	2ralbii 3113	. 2 ⊢ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣))
6	1, 5	bianbi 628	1 ⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) ∧ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ≠ wne 2933  ∀wral 3052   ∩ cin 3889  ∅c0 4274  dom cdm 5626  Rel wrel 5631  [cec 8636   Disj wdisjALTV 38560

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 5232  ax-pr 5372

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 3063  df-rmo 3343  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ec 8640  df-coss 38842  df-cnvrefrel 38948  df-disjALTV 39131

This theorem is referenced by:  disjimeceqim  39145  eldisjs6  39281