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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 718 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)) | |
2 | elun 3222 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐴) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | bitr4i 186 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐵 ∪ 𝐴)) |
4 | 3 | uneqri 3223 | 1 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 698 = wceq 1332 ∈ wcel 1481 ∪ cun 3074 |
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 |
This theorem is referenced by: equncom 3226 uneq2 3229 un12 3239 un23 3240 ssun2 3245 unss2 3252 ssequn2 3254 undir 3331 dif32 3344 undif2ss 3443 uneqdifeqim 3453 prcom 3607 tpass 3627 prprc1 3639 difsnss 3674 suc0 4341 fvun2 5496 fmptpr 5620 fvsnun2 5626 fsnunfv 5629 omv2 6369 phplem2 6755 undifdc 6820 endjusym 6989 fzsuc2 9890 fseq1p1m1 9905 ennnfonelem1 11956 setsslid 12048 exmid1stab 13368 |
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