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Mirrors > Home > MPE Home > Th. List > unass | Structured version Visualization version GIF version |
Description: Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
unass | ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 4106 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∪ 𝐶))) | |
2 | elun 4106 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
3 | 2 | orbi2i 911 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
4 | elun 4106 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
5 | 4 | orbi1i 912 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶)) |
6 | orass 920 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) | |
7 | 5, 6 | bitr2i 275 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∨ 𝑥 ∈ 𝐶)) |
8 | 1, 3, 7 | 3bitrri 297 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐴 ∪ (𝐵 ∪ 𝐶))) |
9 | 8 | uneqri 4109 | 1 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∪ cun 3906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3445 df-un 3913 |
This theorem is referenced by: un12 4125 un23 4126 un4 4127 dfif5 4500 qdass 4712 qdassr 4713 ssunpr 4790 oarec 8505 unfi 9112 domunfican 9260 djuassen 10110 prunioo 13390 ioojoin 13392 fzosplitpr 13673 s4prop 14791 lcmfun 16513 phlstr 17219 prdsvalstr 17326 mreexexlem2d 17517 mreexexlem4d 17519 pwmnd 18739 ordtbas 22527 reconnlem1 24173 lhop 25364 plyun0 25542 ex-un 29254 ex-pw 29259 indifundif 31338 subfacp1lem1 33642 addsasslem2 34316 poimirlem3 36048 poimirlem4 36049 poimirlem16 36061 poimirlem19 36064 dfrcl2 41888 corcltrcl 41953 cotrclrcl 41956 frege131d 41978 |
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