Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  disji2 GIF version

Theorem disji2 3846
 Description: Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑋 ≠ 𝑌, then 𝐶 and 𝐷 are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disji.1 (𝑥 = 𝑋𝐵 = 𝐶)
disji.2 (𝑥 = 𝑌𝐵 = 𝐷)
Assertion
Ref Expression
disji2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disji2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disjnims 3845 . . 3 (Disj 𝑥𝐴 𝐵 → ∀𝑦𝐴𝑧𝐴 (𝑦𝑧 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
2 neeq1 2269 . . . . 5 (𝑦 = 𝑋 → (𝑦𝑧𝑋𝑧))
3 nfcv 2229 . . . . . . . 8 𝑥𝑋
4 nfcv 2229 . . . . . . . 8 𝑥𝐶
5 disji.1 . . . . . . . 8 (𝑥 = 𝑋𝐵 = 𝐶)
63, 4, 5csbhypf 2969 . . . . . . 7 (𝑦 = 𝑋𝑦 / 𝑥𝐵 = 𝐶)
76ineq1d 3203 . . . . . 6 (𝑦 = 𝑋 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = (𝐶𝑧 / 𝑥𝐵))
87eqeq1d 2097 . . . . 5 (𝑦 = 𝑋 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ ↔ (𝐶𝑧 / 𝑥𝐵) = ∅))
92, 8imbi12d 233 . . . 4 (𝑦 = 𝑋 → ((𝑦𝑧 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) ↔ (𝑋𝑧 → (𝐶𝑧 / 𝑥𝐵) = ∅)))
10 neeq2 2270 . . . . 5 (𝑧 = 𝑌 → (𝑋𝑧𝑋𝑌))
11 nfcv 2229 . . . . . . . 8 𝑥𝑌
12 nfcv 2229 . . . . . . . 8 𝑥𝐷
13 disji.2 . . . . . . . 8 (𝑥 = 𝑌𝐵 = 𝐷)
1411, 12, 13csbhypf 2969 . . . . . . 7 (𝑧 = 𝑌𝑧 / 𝑥𝐵 = 𝐷)
1514ineq2d 3204 . . . . . 6 (𝑧 = 𝑌 → (𝐶𝑧 / 𝑥𝐵) = (𝐶𝐷))
1615eqeq1d 2097 . . . . 5 (𝑧 = 𝑌 → ((𝐶𝑧 / 𝑥𝐵) = ∅ ↔ (𝐶𝐷) = ∅))
1710, 16imbi12d 233 . . . 4 (𝑧 = 𝑌 → ((𝑋𝑧 → (𝐶𝑧 / 𝑥𝐵) = ∅) ↔ (𝑋𝑌 → (𝐶𝐷) = ∅)))
189, 17rspc2v 2737 . . 3 ((𝑋𝐴𝑌𝐴) → (∀𝑦𝐴𝑧𝐴 (𝑦𝑧 → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑋𝑌 → (𝐶𝐷) = ∅)))
191, 18mpan9 276 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑌 → (𝐶𝐷) = ∅))
20193impia 1141 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 925   = wceq 1290   ∈ wcel 1439   ≠ wne 2256  ∀wral 2360  ⦋csb 2936   ∩ cin 3001  ∅c0 3289  Disj wdisj 3830 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-in 3008  df-nul 3290  df-disj 3831 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator