Theorem disjne 3478

Description: Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
disjne	⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷)

Proof of Theorem disjne

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disj 3473	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
2		eleq1 2240	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
3	2	notbid 667	. . . . 5 ⊢ (𝑥 = 𝐶 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝐶 ∈ 𝐵))
4	3	rspccva 2842	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵)
5		eleq1a 2249	. . . . 5 ⊢ (𝐷 ∈ 𝐵 → (𝐶 = 𝐷 → 𝐶 ∈ 𝐵))
6	5	necon3bd 2390	. . . 4 ⊢ (𝐷 ∈ 𝐵 → (¬ 𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐷))
7	4, 6	syl5com 29	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → 𝐶 ≠ 𝐷))
8	1, 7	sylanb 284	. 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → 𝐶 ≠ 𝐷))
9	8	3impia 1200	1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455   ∩ cin 3130  ∅c0 3424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2741  df-dif 3133  df-in 3137  df-nul 3425

This theorem is referenced by: (None)