Theorem undif2ss 3438

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 3437	. 2 ⊢ ((𝐵 ∖ 𝐴) ∪ 𝐴) ⊆ (𝐵 ∪ 𝐴)
2		uncom 3220	. 2 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ 𝐴)
3		uncom 3220	. 2 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
4	1, 2, 3	3sstr4i 3138	1 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∖ cdif 3068   ∪ cun 3069   ⊆ wss 3071

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084

This theorem is referenced by: (None)