Theorem eldisjim3 39197

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 1143	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → 𝐵 ∈ 𝐴)
2		simp2 1144	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → 𝐶 ∈ 𝐴)
3		eleq1 2829	. . . . . 6 ⊢ (𝑢 = 𝐵 → (𝑢 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4		eleq1 2829	. . . . . 6 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
5	3, 4	bi2anan9 645	. . . . 5 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)))
6		ineq12 4147	. . . . . . 7 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑢 ∩ 𝑣) = (𝐵 ∩ 𝐶))
7	6	neeq1d 2995	. . . . . 6 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → ((𝑢 ∩ 𝑣) ≠ ∅ ↔ (𝐵 ∩ 𝐶) ≠ ∅))
8		eqeq12 2758	. . . . . 6 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑢 = 𝑣 ↔ 𝐵 = 𝐶))
9	7, 8	imbi12d 346	. . . . 5 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣) ↔ ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))
10	5, 9	imbi12d 346	. . . 4 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)) ↔ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶))))
11		dfeldisj5a 39196	. . . . . 6 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
12		rsp2 3258	. . . . . 6 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)))
13	11, 12	sylbi 219	. . . . 5 ⊢ ( ElDisj 𝐴 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)))
14	13	3ad2ant3 1142	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)))
15	1, 2, 10, 14	vtocl2d 3509	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))
16	15	3expia 1128	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ( ElDisj 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶))))
17	16	pm2.43b 55	1 ⊢ ( ElDisj 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055   ∩ cin 3884  ∅c0 4264   ElDisj weldisj 38603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8639  df-coss 38883  df-cnvrefrel 38989  df-disjALTV 39172  df-eldisj 39174

This theorem is referenced by:  eldisjdmqsim2  39198  eldisjdmqsim  39199  suceldisj  39200  rnqmapeleldisjsim  39244