Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ssequn2 | GIF version |
Description: A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.) |
Ref | Expression |
---|---|
ssequn2 | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn1 3251 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) | |
2 | uncom 3225 | . . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | |
3 | 2 | eqeq1i 2148 | . 2 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
4 | 1, 3 | bitri 183 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1332 ∪ cun 3074 ⊆ wss 3076 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 |
This theorem is referenced by: unabs 3312 pwssunim 4214 pwundifss 4215 oneluni 4361 relresfld 5076 relcoi1 5078 fsnunf 5628 unsnfidcel 6817 fidcenumlemr 6851 exmidfodomrlemim 7074 ennnfonelemhf1o 11962 |
Copyright terms: Public domain | W3C validator |