Theorem eldisjim3 39938

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 1154	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → 𝐵 ∈ 𝐴)
2		simp2 1155	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → 𝐶 ∈ 𝐴)
3		eleq1 2852	. . . . . 6 ⊢ (𝑢 = 𝐵 → (𝑢 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4		eleq1 2852	. . . . . 6 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
5	3, 4	bi2anan9 650	. . . . 5 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)))
6		ineq12 4115	. . . . . . 7 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑢 ∩ 𝑣) = (𝐵 ∩ 𝐶))
7	6	neeq1d 3018	. . . . . 6 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → ((𝑢 ∩ 𝑣) ≠ ∅ ↔ (𝐵 ∩ 𝐶) ≠ ∅))
8		eqeq12 2780	. . . . . 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 39937	. . . . . 6 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
12		rsp2 3283	. . . . . 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 1153	. . . 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 1139	. 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 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ∩ cin 4092  ∅c0 4281   ElDisj weldisj 39344

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8744  df-coss 39624  df-cnvrefrel 39730  df-disjALTV 39913  df-eldisj 39915

This theorem is used by:  eldisjdmqsim2  39939  eldisjdmqsim  39940  suceldisj  39941  rnqmapeleldisjsim  39985