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Mirrors > Home > MPE Home > Th. List > unexg | Structured version Visualization version GIF version |
Description: A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) |
Ref | Expression |
---|---|
unexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3364 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | elex 3364 | . 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
3 | unexb 7105 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) | |
4 | 3 | biimpi 206 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) |
5 | 1, 2, 4 | syl2an 583 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 Vcvv 3351 ∪ cun 3721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-rex 3067 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-sn 4317 df-pr 4319 df-uni 4575 |
This theorem is referenced by: xpexg 7107 difex2 7116 difsnexi 7117 eldifpw 7123 ordunpr 7173 soex 7256 fnse 7445 suppun 7466 tposexg 7518 wfrlem15 7582 tfrlem12 7638 tfrlem16 7642 ralxpmap 8061 undifixp 8098 undom 8204 domunsncan 8216 domssex2 8276 domssex 8277 mapunen 8285 fsuppunbi 8452 elfiun 8492 brwdom2 8634 unwdomg 8645 djuex 8935 djuexALT 8948 alephprc 9122 cdadom3 9212 infunabs 9231 fin23lem11 9341 axdc2lem 9472 ttukeylem1 9533 fpwwe2lem13 9666 wunex2 9762 wuncval2 9771 hashunx 13377 hashf1lem1 13441 trclexlem 13943 trclun 13963 relexp0g 13970 relexpsucnnr 13973 isstruct2 16074 setsvalg 16094 setsid 16121 yonffth 17132 dmdprdsplit2 18653 basdif0 20978 fiuncmp 21428 refun0 21539 ptbasfi 21605 dfac14lem 21641 ptrescn 21663 xkoptsub 21678 filconn 21907 isufil2 21932 ufileu 21943 filufint 21944 fmfnfmlem4 21981 fmfnfm 21982 fclsfnflim 22051 flimfnfcls 22052 ptcmplem1 22076 elply2 24172 plyss 24175 wlkp1lem4 26808 resf1o 29845 locfinref 30248 esumsplit 30455 esumpad2 30458 sseqval 30790 bnj1149 31201 ssltun1 32252 ssltun2 32253 altxpexg 32422 hfun 32622 refssfne 32690 topjoin 32697 bj-2uplex 33341 cnfinltrel 33578 ptrest 33741 poimirlem3 33745 paddval 35606 elrfi 37783 elmapresaun 37860 rclexi 38448 rtrclexlem 38449 trclubgNEW 38451 clcnvlem 38456 cnvrcl0 38458 dfrtrcl5 38462 iunrelexp0 38520 relexpmulg 38528 relexp01min 38531 relexpxpmin 38535 brtrclfv2 38545 sge0resplit 41140 sge0split 41143 setsv 41876 setrec1lem4 42965 |
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