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Theorem unidif 4847
 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 4846 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 𝐴 (𝐴𝐵))
2 difss 4083 . . . 4 (𝐴𝐵) ⊆ 𝐴
32unissi 4822 . . 3 (𝐴𝐵) ⊆ 𝐴
41, 3jctil 523 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 → ( (𝐴𝐵) ⊆ 𝐴 𝐴 (𝐴𝐵)))
5 eqss 3957 . 2 ( (𝐴𝐵) = 𝐴 ↔ ( (𝐴𝐵) ⊆ 𝐴 𝐴 (𝐴𝐵)))
64, 5sylibr 237 1 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∀wral 3130  ∃wrex 3131   ∖ cdif 3905   ⊆ wss 3908  ∪ cuni 4813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-uni 4814 This theorem is referenced by:  ordunidif  6217
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