Theorem eldisjim3 39492

Description: ElDisj elimination (two chosen elements). Standard specialization lemma: from ElDisj 𝐴 infer the disjointness condition for two specific elements. (Contributed by Peter Mazsa, 6-Feb-2026.)

Assertion

Ref	Expression
eldisjim3	⊢ ( ElDisj 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))

Proof of Theorem eldisjim3

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1153	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → 𝐵 ∈ 𝐴)
2		simp2 1154	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → 𝐶 ∈ 𝐴)
3		eleq1 2850	. . . . . 6 ⊢ (𝑢 = 𝐵 → (𝑢 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4		eleq1 2850	. . . . . 6 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
5	3, 4	bi2anan9 649	. . . . 5 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)))
6		ineq12 4167	. . . . . . 7 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑢 ∩ 𝑣) = (𝐵 ∩ 𝐶))
7	6	neeq1d 3016	. . . . . 6 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → ((𝑢 ∩ 𝑣) ≠ ∅ ↔ (𝐵 ∩ 𝐶) ≠ ∅))
8		eqeq12 2779	. . . . . 6 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑢 = 𝑣 ↔ 𝐵 = 𝐶))
9	7, 8	imbi12d 347	. . . . 5 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣) ↔ ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))
10	5, 9	imbi12d 347	. . . 4 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)) ↔ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶))))
11		dfeldisj5a 39491	. . . . . 6 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
12		rsp2 3281	. . . . . 6 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)))
13	11, 12	sylbi 220	. . . . 5 ⊢ ( ElDisj 𝐴 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)))
14	13	3ad2ant3 1152	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)))
15	1, 2, 10, 14	vtocl2d 3527	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))
16	15	3expia 1138	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ( ElDisj 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶))))
17	16	pm2.43b 56	1 ⊢ ( ElDisj 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078   ∩ cin 3903  ∅c0 4285   ElDisj weldisj 38898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-eprel 5560  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ec 8694  df-coss 39178  df-cnvrefrel 39284  df-disjALTV 39467  df-eldisj 39469

This theorem is used by:  eldisjdmqsim2  39493  eldisjdmqsim  39494  suceldisj  39495  rnqmapeleldisjsim  39539