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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 3715 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∪ 𝐶))) | |
2 | elun 3715 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
3 | 2 | orbi2i 540 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
4 | elun 3715 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
5 | 4 | orbi1i 541 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶)) |
6 | orass 545 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) | |
7 | 5, 6 | bitr2i 264 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∨ 𝑥 ∈ 𝐶)) |
8 | 1, 3, 7 | 3bitrri 286 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∨ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐴 ∪ (𝐵 ∪ 𝐶))) |
9 | 8 | uneqri 3717 | 1 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 |
This theorem is referenced by: un12 3733 un23 3734 un4 3735 dfif5 4052 qdass 4232 qdassr 4233 ssunpr 4305 oarec 7529 domunfican 8118 cdaassen 8887 prunioo 12172 ioojoin 12174 fzosplitprm1 12443 s4prop 13505 lcmfun 15196 strlemor2 15797 strlemor3 15798 phlstr 15857 prdsvalstr 15936 mreexexlem2d 16128 mreexexlem4d 16130 ordtbas 20806 reconnlem1 22437 lhop 23583 plyun0 23757 ex-un 26673 ex-pw 26678 indifundif 28740 subfacp1lem1 30415 poimirlem3 32582 poimirlem4 32583 poimirlem16 32595 poimirlem19 32598 dfrcl2 36985 corcltrcl 37050 cotrclrcl 37053 frege131d 37075 fzosplitpr 40362 |
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