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Mirrors > Home > MPE Home > Th. List > undifr | Structured version Visualization version GIF version |
Description: Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.) (Proof shortened by SN, 11-Mar-2025.) |
Ref | Expression |
---|---|
undifr | ⊢ (𝐴 ⊆ 𝐵 ↔ ((𝐵 ∖ 𝐴) ∪ 𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn2 4199 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) | |
2 | undif1 4482 | . . 3 ⊢ ((𝐵 ∖ 𝐴) ∪ 𝐴) = (𝐵 ∪ 𝐴) | |
3 | 2 | eqeq1i 2740 | . 2 ⊢ (((𝐵 ∖ 𝐴) ∪ 𝐴) = 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
4 | 1, 3 | bitr4i 278 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ ((𝐵 ∖ 𝐴) ∪ 𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 |
This theorem is referenced by: difsnid 4815 f1ofvswap 7326 ralxpmap 8935 psdmullem 22187 psdmul 22188 tocyc01 33121 rprmdvdsprod 33542 aks6d1c5lem3 42119 selvvvval 42572 evlselvlem 42573 evlselv 42574 isubgr3stgrlem3 47871 |
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