Theorem undif2ss 3570

Description: Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
undif2ss	⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ (𝐴 ∪ 𝐵)

Proof of Theorem undif2ss

Step	Hyp	Ref	Expression
1		undif1ss 3569	. 2 ⊢ ((𝐵 ∖ 𝐴) ∪ 𝐴) ⊆ (𝐵 ∪ 𝐴)
2		uncom 3351	. 2 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ 𝐴)
3		uncom 3351	. 2 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
4	1, 2, 3	3sstr4i 3268	1 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∖ cdif 3197   ∪ cun 3198   ⊆ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213

This theorem is referenced by: (None)