Theorem disjne 3491

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 3486	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
2		eleq1 2252	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
3	2	notbid 668	. . . . 5 ⊢ (𝑥 = 𝐶 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝐶 ∈ 𝐵))
4	3	rspccva 2855	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵)
5		eleq1a 2261	. . . . 5 ⊢ (𝐷 ∈ 𝐵 → (𝐶 = 𝐷 → 𝐶 ∈ 𝐵))
6	5	necon3bd 2403	. . . 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 1202	1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2160   ≠ wne 2360  ∀wral 2468   ∩ cin 3143  ∅c0 3437

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-nul 3438

This theorem is referenced by: (None)