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Mirrors > Home > MPE Home > Th. List > uniex | Structured version Visualization version GIF version |
Description: The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 3492), then the union of 𝐴 is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.) |
Ref | Expression |
---|---|
uniex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
uniex | ⊢ ∪ 𝐴 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | uniexg 7759 | . 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∪ 𝐴 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3478 ∪ cuni 4912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-uni 4913 |
This theorem is referenced by: unexOLD 7764 iunpw 7790 elxp4 7945 elxp5 7946 1stval 8015 2ndval 8016 fo1st 8033 fo2nd 8034 cnvf1o 8135 brtpos2 8256 naddcllem 8713 ixpsnf1o 8977 dffi3 9469 cnfcom2 9740 cnfcom3lem 9741 cnfcom3 9742 ttrclse 9765 trcl 9766 rankc2 9909 rankxpl 9913 rankxpsuc 9920 acnlem 10086 dfac2a 10168 fin23lem14 10371 fin23lem16 10373 fin23lem17 10376 fin23lem38 10387 fin23lem39 10388 itunisuc 10457 axdc3lem2 10489 axcclem 10495 ac5b 10516 ttukey 10556 wunex2 10776 wuncval2 10785 intgru 10852 pnfex 11312 prdsvallem 17501 prdsval 17502 prdsds 17511 wunfunc 17952 wunfuncOLD 17953 wunnat 18011 wunnatOLD 18012 arwval 18097 catcfuccl 18173 catcfucclOLD 18174 catcxpccl 18263 catcxpcclOLD 18264 zrhval 21536 mreclatdemoBAD 23120 ptbasin2 23602 ptbasfi 23605 dfac14 23642 ptcmplem2 24077 ptcmplem3 24078 ptcmp 24082 cnextfvval 24089 cnextcn 24091 minveclem4a 25478 oldf 27911 madefi 27965 precsexlem10 28255 xrge0tsmsbi 33049 dimval 33628 dimvalfi 33629 locfinreflem 33801 pstmfval 33857 pstmxmet 33858 esumex 34010 msrval 35523 dfrdg2 35777 fvbigcup 35884 ctbssinf 37389 ptrest 37606 heiborlem1 37798 heiborlem3 37800 heibor 37808 dicval 41159 prjcrvfval 42618 aomclem1 43043 dfac21 43055 ntrrn 44112 ntrf 44113 dssmapntrcls 44118 fourierdlem70 46132 caragendifcl 46470 cnfsmf 46696 setrec1lem3 48920 setrec2fun 48923 |
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