Theorem eldisjim3 39319

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 1150	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → 𝐵 ∈ 𝐴)
2		simp2 1151	. . . 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 647	. . . . 5 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)))
6		ineq12 4169	. . . . . . 7 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑢 ∩ 𝑣) = (𝐵 ∩ 𝐶))
7	6	neeq1d 3018	. . . . . 6 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → ((𝑢 ∩ 𝑣) ≠ ∅ ↔ (𝐵 ∩ 𝐶) ≠ ∅))
8		eqeq12 2781	. . . . . 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 39318	. . . . . 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 219	. . . . 5 ⊢ ( ElDisj 𝐴 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)))
14	13	3ad2ant3 1149	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣)))
15	1, 2, 10, 14	vtocl2d 3530	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ ElDisj 𝐴) → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))
16	15	3expia 1135	. 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 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078   ∩ cin 3905  ∅c0 4287   ElDisj weldisj 38725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-id 5544  df-eprel 5549  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682  df-coss 39005  df-cnvrefrel 39111  df-disjALTV 39294  df-eldisj 39296

This theorem is referenced by:  eldisjdmqsim2  39320  eldisjdmqsim  39321  suceldisj  39322  rnqmapeleldisjsim  39366