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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 2692 | . 2 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 → 𝐴 ∈ V) | |
2 | elex 2692 | . . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
3 | 2 | rexlimivw 2543 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 → 𝐴 ∈ V) |
4 | eleq1 2200 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
5 | 4 | rexbidv 2436 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
6 | df-iun 3810 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶} | |
7 | 5, 6 | elab2g 2826 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
8 | 1, 3, 7 | pm5.21nii 693 | 1 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃wrex 2415 Vcvv 2681 ∪ ciun 3808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-iun 3810 |
This theorem is referenced by: iuncom 3814 iuncom4 3815 iunconstm 3816 iuniin 3818 iunss1 3819 ss2iun 3823 dfiun2g 3840 ssiun 3850 ssiun2 3851 iunab 3854 iun0 3864 0iun 3865 iunn0m 3868 iunin2 3871 iundif2ss 3873 iindif2m 3875 iunxsng 3883 iunxsngf 3885 iunun 3886 iunxun 3887 iunxiun 3889 iunpwss 3899 disjiun 3919 triun 4034 iunpw 4396 xpiundi 4592 xpiundir 4593 iunxpf 4682 cnvuni 4720 dmiun 4743 dmuni 4744 rniun 4944 dfco2 5033 dfco2a 5034 coiun 5043 fun11iun 5381 imaiun 5654 eluniimadm 5659 opabex3d 6012 opabex3 6013 smoiun 6191 tfrlemi14d 6223 tfr1onlemres 6239 tfrcllemres 6252 fsum2dlemstep 11196 fisumcom2 11200 fsumiun 11239 ennnfonelemrn 11921 ennnfonelemdm 11922 ctiunctlemf 11940 ctiunctlemfo 11941 |
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