Theorem eldisjdmqsim2 39183

Description: ElDisj of quotient implies coset-disjointness (domain form). Converts element-disjointness of the quotient carrier into a usable "cosets don't overlap unless equal" rule. (Contributed by Peter Mazsa, 10-Feb-2026.)

Assertion

Ref	Expression
eldisjdmqsim2	⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))

Proof of Theorem eldisjdmqsim2

Step	Hyp	Ref	Expression
1		eldisjim3 39182	. . 3 ⊢ ( ElDisj (dom 𝑅 / 𝑅) → (([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))
2		eceldmqs 8724	. . . . 5 ⊢ (𝑅 ∈ Rels → ([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑢 ∈ dom 𝑅))
3		eceldmqs 8724	. . . . 5 ⊢ (𝑅 ∈ Rels → ([𝑣]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑣 ∈ dom 𝑅))
4	2, 3	anbi12d 638	. . . 4 ⊢ (𝑅 ∈ Rels → (([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)))
5	4	imbi1d 342	. . 3 ⊢ (𝑅 ∈ Rels → ((([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)) ↔ ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅))))
6	1, 5	imbitrid 245	. 2 ⊢ (𝑅 ∈ Rels → ( ElDisj (dom 𝑅 / 𝑅) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅))))
7	6	impcom 408	1 ⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   ∩ cin 3882  ∅c0 4261  dom cdm 5618  [cec 8631   / cqs 8632   Rels crels 38552   ElDisj weldisj 38588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-eprel 5518  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ec 8635  df-qs 8639  df-coss 38868  df-cnvrefrel 38974  df-disjALTV 39157  df-eldisj 39159

This theorem is referenced by: (None)