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Mirrors > Home > MPE Home > Th. List > nfiu1 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.) |
Ref | Expression |
---|---|
nfiu1 | ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 4522 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
2 | nfre1 3005 | . . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfab 2769 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
4 | 1, 3 | nfcxfr 2762 | 1 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 {cab 2608 Ⅎwnfc 2751 ∃wrex 2913 ∪ ciun 4520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-iun 4522 |
This theorem is referenced by: ssiun2s 4564 disjxiun 4649 disjxiunOLD 4650 triun 4766 iunopeqop 4981 eliunxp 5259 opeliunxp2 5260 opeliunxp2f 7336 ixpf 7930 ixpiunwdom 8496 r1val1 8649 rankuni2b 8716 rankval4 8730 cplem2 8753 ac6num 9301 iunfo 9361 iundom2g 9362 inar1 9597 tskuni 9605 gsum2d2lem 18372 gsum2d2 18373 gsumcom2 18374 iunconn 21231 ptclsg 21418 cnextfvval 21869 ssiun2sf 29378 aciunf1lem 29462 fsumiunle 29575 esum2dlem 30154 esum2d 30155 esumiun 30156 sigapildsys 30225 bnj958 31010 bnj1000 31011 bnj981 31020 bnj1398 31102 bnj1408 31104 iunconnlem2 39171 iunmapss 39407 iunmapsn 39409 allbutfi 39616 fsumiunss 39807 dvnprodlem1 40161 dvnprodlem2 40162 sge0iunmptlemfi 40630 sge0iunmptlemre 40632 sge0iunmpt 40635 iundjiun 40677 voliunsge0lem 40689 caratheodorylem2 40741 smflimmpt 41016 smflimsuplem7 41032 eliunxp2 42112 |
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