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Mirrors > Home > ILE Home > Th. List > undif2ss | GIF version |
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.) |
Ref | Expression |
---|---|
undif2ss | ⊢ (A ∪ (B ∖ A)) ⊆ (A ∪ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undif1ss 3292 | . 2 ⊢ ((B ∖ A) ∪ A) ⊆ (B ∪ A) | |
2 | uncom 3081 | . 2 ⊢ (A ∪ (B ∖ A)) = ((B ∖ A) ∪ A) | |
3 | uncom 3081 | . 2 ⊢ (A ∪ B) = (B ∪ A) | |
4 | 1, 2, 3 | 3sstr4i 2978 | 1 ⊢ (A ∪ (B ∖ A)) ⊆ (A ∪ B) |
Colors of variables: wff set class |
Syntax hints: ∖ cdif 2908 ∪ cun 2909 ⊆ wss 2911 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 |
This theorem is referenced by: (None) |
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