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Mirrors > Home > MPE Home > Th. List > eluni2 | Structured version Visualization version GIF version |
Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.) |
Ref | Expression |
---|---|
eluni2 | ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1774 | . 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
2 | eluni 4375 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) | |
3 | df-rex 2902 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
4 | 1, 2, 3 | 3bitr4i 291 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 ∪ cuni 4372 |
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-rex 2902 df-v 3175 df-uni 4373 |
This theorem is referenced by: uni0b 4399 intssuni 4434 iuncom4 4464 inuni 4753 cnvuni 5231 chfnrn 6236 ssorduni 6877 unon 6923 limuni3 6944 onfununi 7325 oarec 7529 uniinqs 7714 fissuni 8154 finsschain 8156 r1sdom 8520 rankuni2b 8599 cflm 8955 coflim 8966 axdc3lem2 9156 fpwwe2lem12 9342 uniwun 9441 tskr1om2 9469 tskuni 9484 axgroth3 9532 inaprc 9537 tskmval 9540 tskmcl 9542 suplem1pr 9753 lbsextlem2 18980 lbsextlem3 18981 isbasis3g 20564 eltg2b 20574 tgcl 20584 ppttop 20621 epttop 20623 neiptoptop 20745 tgcmp 21014 locfincmp 21139 dissnref 21141 comppfsc 21145 1stckgenlem 21166 txuni2 21178 txcmplem1 21254 tgqtop 21325 filuni 21499 alexsubALTlem4 21664 ptcmplem3 21668 ptcmplem4 21669 utoptop 21848 icccmplem1 22433 icccmplem3 22435 cnheibor 22562 bndth 22565 lebnumlem1 22568 bcthlem4 22932 ovolicc2lem5 23096 dyadmbllem 23173 itg2gt0 23333 rexunirn 28715 unipreima 28826 acunirnmpt2 28842 acunirnmpt2f 28843 reff 29234 locfinreflem 29235 cmpcref 29245 ddemeas 29626 dya2iocuni 29672 bnj1379 30155 cvmsss2 30510 cvmseu 30512 untuni 30840 dfon2lem3 30934 dfon2lem7 30938 dfon2lem8 30939 brbigcup 31175 neibastop1 31524 neibastop2lem 31525 heicant 32614 mblfinlem1 32616 cover2 32678 heiborlem9 32788 unichnidl 33000 prtlem16 33172 prter2 33184 prter3 33185 restuni3 38333 disjinfi 38375 cncfuni 38772 intsaluni 39223 salgencntex 39237 |
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