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Mirrors > Home > ILE Home > Th. List > uncom | GIF version |
Description: Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
uncom | ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orcom 728 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)) | |
2 | elun 3276 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐴) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | bitr4i 187 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐵 ∪ 𝐴)) |
4 | 3 | uneqri 3277 | 1 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∪ cun 3127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 |
This theorem is referenced by: equncom 3280 uneq2 3283 un12 3293 un23 3294 ssun2 3299 unss2 3306 ssequn2 3308 undir 3385 dif32 3398 undif2ss 3498 uneqdifeqim 3508 prcom 3667 tpass 3687 prprc1 3699 difsnss 3737 suc0 4407 fvun2 5578 fmptpr 5703 fvsnun2 5709 fsnunfv 5712 omv2 6459 phplem2 6846 undifdc 6916 endjusym 7088 fzsuc2 10052 fseq1p1m1 10067 ennnfonelem1 12378 setsslid 12482 exmid1stab 14372 |
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