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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 717 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)) | |
2 | elun 3212 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐴) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | bitr4i 186 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐵 ∪ 𝐴)) |
4 | 3 | uneqri 3213 | 1 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 697 = wceq 1331 ∈ wcel 1480 ∪ cun 3064 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 |
This theorem is referenced by: equncom 3216 uneq2 3219 un12 3229 un23 3230 ssun2 3235 unss2 3242 ssequn2 3244 undir 3321 dif32 3334 undif2ss 3433 uneqdifeqim 3443 prcom 3594 tpass 3614 prprc1 3626 difsnss 3661 suc0 4328 fvun2 5481 fmptpr 5605 fvsnun2 5611 fsnunfv 5614 omv2 6354 phplem2 6740 undifdc 6805 endjusym 6974 fzsuc2 9852 fseq1p1m1 9867 ennnfonelem1 11909 setsslid 11998 exmid1stab 13184 |
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