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Theorem unidif 3691
 Description: If the difference 𝐴 ∖ 𝐵 contains the largest members of 𝐴, then the union of the difference is the union of 𝐴. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
unidif (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem unidif
StepHypRef Expression
1 uniss2 3690 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 𝐴 (𝐴𝐵))
2 difss 3127 . . . 4 (𝐴𝐵) ⊆ 𝐴
32unissi 3682 . . 3 (𝐴𝐵) ⊆ 𝐴
41, 3jctil 306 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 → ( (𝐴𝐵) ⊆ 𝐴 𝐴 (𝐴𝐵)))
5 eqss 3041 . 2 ( (𝐴𝐵) = 𝐴 ↔ ( (𝐴𝐵) ⊆ 𝐴 𝐴 (𝐴𝐵)))
64, 5sylibr 133 1 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1290  ∀wral 2360  ∃wrex 2361   ∖ cdif 2997   ⊆ wss 3000  ∪ cuni 3659 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-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-uni 3660 This theorem is referenced by: (None)
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