Theorem undif1ss 3588

Description: Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
undif1ss	⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐴 ∪ 𝐵)

Proof of Theorem undif1ss

Step	Hyp	Ref	Expression
1		difss 3349	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		unss1 3392	. 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐴 ∪ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∖ cdif 3211   ∪ cun 3212   ⊆ wss 3214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227

This theorem is referenced by:  undif2ss  3589  pwundifss  4411