Theorem undif2ss 3296

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 3295	. 2 ⊢ ((𝐵 ∖ 𝐴) ∪ 𝐴) ⊆ (𝐵 ∪ 𝐴)
2		uncom 3084	. 2 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ 𝐴)
3		uncom 3084	. 2 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
4	1, 2, 3	3sstr4i 2981	1 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∖ cdif 2911   ∪ cun 2912   ⊆ wss 2914

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928

This theorem is referenced by: (None)