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Mirrors > Home > ILE Home > Th. List > eliun | GIF version |
Description: Membership in indexed union. (Contributed by NM, 3-Sep-2003.) |
Ref | Expression |
---|---|
eliun | ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2750 | . 2 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 → 𝐴 ∈ V) | |
2 | elex 2750 | . . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
3 | 2 | rexlimivw 2590 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 → 𝐴 ∈ V) |
4 | eleq1 2240 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
5 | 4 | rexbidv 2478 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
6 | df-iun 3890 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶} | |
7 | 5, 6 | elab2g 2886 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
8 | 1, 3, 7 | pm5.21nii 704 | 1 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 Vcvv 2739 ∪ ciun 3888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-iun 3890 |
This theorem is referenced by: iuncom 3894 iuncom4 3895 iunconstm 3896 iuniin 3898 iunss1 3899 ss2iun 3903 dfiun2g 3920 ssiun 3930 ssiun2 3931 iunab 3935 iun0 3945 0iun 3946 iunn0m 3949 iunin2 3952 iundif2ss 3954 iindif2m 3956 iunxsng 3964 iunxsngf 3966 iunun 3967 iunxun 3968 iunxiun 3970 iunpwss 3980 disjiun 4000 triun 4116 iunpw 4482 xpiundi 4686 xpiundir 4687 iunxpf 4777 cnvuni 4815 dmiun 4838 dmuni 4839 rniun 5041 dfco2 5130 dfco2a 5131 coiun 5140 fun11iun 5484 imaiun 5764 eluniimadm 5769 opabex3d 6125 opabex3 6126 smoiun 6305 tfrlemi14d 6337 tfr1onlemres 6353 tfrcllemres 6366 fsum2dlemstep 11445 fisumcom2 11449 fsumiun 11488 fprod2dlemstep 11633 fprodcom2fi 11637 ennnfonelemrn 12423 ennnfonelemdm 12424 ctiunctlemf 12442 ctiunctlemfo 12443 imasaddfnlemg 12741 lssats2 13539 |
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