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Mirrors > Home > NFE 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 2867 | . 2 ⊢ (A ∈ ∪x ∈ B C → A ∈ V) | |
2 | elex 2867 | . . 3 ⊢ (A ∈ C → A ∈ V) | |
3 | 2 | rexlimivw 2734 | . 2 ⊢ (∃x ∈ B A ∈ C → A ∈ V) |
4 | eleq1 2413 | . . . 4 ⊢ (y = A → (y ∈ C ↔ A ∈ C)) | |
5 | 4 | rexbidv 2635 | . . 3 ⊢ (y = A → (∃x ∈ B y ∈ C ↔ ∃x ∈ B A ∈ C)) |
6 | df-iun 3971 | . . 3 ⊢ ∪x ∈ B C = {y ∣ ∃x ∈ B y ∈ C} | |
7 | 5, 6 | elab2g 2987 | . 2 ⊢ (A ∈ V → (A ∈ ∪x ∈ B C ↔ ∃x ∈ B A ∈ C)) |
8 | 1, 3, 7 | pm5.21nii 342 | 1 ⊢ (A ∈ ∪x ∈ B C ↔ ∃x ∈ B A ∈ C) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 Vcvv 2859 ∪ciun 3969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-iun 3971 |
This theorem is referenced by: iuncom 3975 iuncom4 3976 iunconst 3977 iuniin 3979 iunss1 3980 ss2iun 3984 dfiun2g 3999 ssiun 4008 ssiun2 4009 iunab 4012 iun0 4022 0iun 4023 iunn0 4026 iunin2 4030 iundif2 4033 iindif2 4035 iunxsng 4044 iunun 4046 iunxun 4047 iunxiun 4048 iunpwss 4055 xpiundi 4817 xpiundir 4818 iunxpf 4829 cnvuni 4895 dmuni 4914 rnuni 5038 dfco2 5080 dfco2a 5081 coiun 5090 fun11iun 5305 imaiun 5464 eluniima 5469 |
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