Theorem phplem1 3403

Description: Lemma for Pigeonhole Principle. This just says that if we remove an element of a set then put it back in, we end up with the original set.

Assertion

Ref	Expression
phplem1	⊢ (B ∈ A → ({B} ∪ (A ∖ {B})) = A)

Proof of Theorem phplem1

Step	Hyp	Ref	Expression
1		snssi 1851	. 2 ⊢ (B ∈ A → {B} ⊆ A)
2		ssundif 1764	. 2 ⊢ ({B} ⊆ A ↔ ({B} ∪ (A ∖ {B})) = A)
3	1, 2	sylib 173	1 ⊢ (B ∈ A → ({B} ∪ (A ∖ {B})) = A)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ∪ cun 1485   ⊆ wss 1487  {csn 1808

This theorem is referenced by:  phplem3 3405

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811

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