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Theorem unidif 3826
Description: If the difference 𝐴𝐵 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 3825 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 𝐴 (𝐴𝐵))
2 difss 3253 . . . 4 (𝐴𝐵) ⊆ 𝐴
32unissi 3817 . . 3 (𝐴𝐵) ⊆ 𝐴
41, 3jctil 310 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 → ( (𝐴𝐵) ⊆ 𝐴 𝐴 (𝐴𝐵)))
5 eqss 3162 . 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 1348  wral 2448  wrex 2449  cdif 3118  wss 3121   cuni 3794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-uni 3795
This theorem is referenced by: (None)
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