Theorem disjne 3518

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 3513	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
2		eleq1 2269	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
3	2	notbid 669	. . . . 5 ⊢ (𝑥 = 𝐶 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝐶 ∈ 𝐵))
4	3	rspccva 2880	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵)
5		eleq1a 2278	. . . . 5 ⊢ (𝐷 ∈ 𝐵 → (𝐶 = 𝐷 → 𝐶 ∈ 𝐵))
6	5	necon3bd 2420	. . . 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 1203	1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2177   ≠ wne 2377  ∀wral 2485   ∩ cin 3169  ∅c0 3464

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-v 2775  df-dif 3172  df-in 3176  df-nul 3465

This theorem is referenced by: (None)