Theorem eldisjim3 39667

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 2848	. . . . . 6 ⊢ (𝑢 = 𝐵 → (𝑢 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4		eleq1 2848	. . . . . 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 4160	. . . . . . 7 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑢 ∩ 𝑣) = (𝐵 ∩ 𝐶))
7	6	neeq1d 3014	. . . . . 6 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → ((𝑢 ∩ 𝑣) ≠ ∅ ↔ (𝐵 ∩ 𝐶) ≠ ∅))
8		eqeq12 2777	. . . . . 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 39666	. . . . . 6 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
12		rsp2 3279	. . . . . 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 3523	. . 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 2145   ≠ wne 2955  ∀wral 3076   ∩ cin 3897  ∅c0 4278   ElDisj weldisj 39073

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5542  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8697  df-coss 39353  df-cnvrefrel 39459  df-disjALTV 39642  df-eldisj 39644

This theorem is used by:  eldisjdmqsim2  39668  eldisjdmqsim  39669  suceldisj  39670  rnqmapeleldisjsim  39714