Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3478 | . 2 ⊢ ((𝐵 ∖ 𝐴) ∪ 𝐴) ⊆ (𝐵 ∪ 𝐴) | |
2 | uncom 3261 | . 2 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ 𝐴) | |
3 | uncom 3261 | . 2 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | |
4 | 1, 2, 3 | 3sstr4i 3178 | 1 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∖ cdif 3108 ∪ cun 3109 ⊆ wss 3111 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |