![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eluni | Structured version Visualization version GIF version |
Description: Membership in class union. (Contributed by NM, 22-May-1994.) |
Ref | Expression |
---|---|
eluni | ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3498 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V) | |
2 | elex 3498 | . . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ V) | |
3 | 2 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V) |
4 | 3 | exlimiv 1927 | . 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V) |
5 | eleq1 2826 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
6 | 5 | anbi1d 631 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
7 | 6 | exbidv 1918 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
8 | df-uni 4912 | . . 3 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)} | |
9 | 7, 8 | elab2g 3682 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
10 | 1, 4, 9 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 ∪ cuni 4911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-uni 4912 |
This theorem is referenced by: eluni2 4915 elunii 4916 uniss 4919 eluniab 4925 uniun 4934 uniin 4935 unissb 4943 unissbOLD 4944 dfiun2g 5034 dftr2 5266 unipw 5460 dmuni 5927 iotanul2 6532 fununi 6642 elunirn 7270 uniex2 7756 uniuni 7780 mpoxopxnop0 8238 fprresex 8333 wfrfunOLD 8357 wfrlem17OLD 8363 tfrlem7 8421 tfrlem9a 8424 inf2 9660 inf3lem2 9666 rankwflemb 9830 cardprclem 10016 carduni 10018 iunfictbso 10151 kmlem3 10190 kmlem4 10191 cfub 10286 isf34lem4 10414 grothtsk 10872 suplem1pr 11089 toprntopon 22946 isbasis2g 22970 tgval2 22978 ntreq0 23100 cmpsublem 23422 cmpsub 23423 cmpcld 23425 is1stc2 23465 alexsubALTlem3 24072 alexsubALT 24074 elold 27922 fnessref 36339 bj-restuni 37079 difunieq 37356 ismnushort 44296 truniALT 44538 truniALTVD 44875 unisnALT 44923 elunif 44953 ssfiunibd 45259 stoweidlem27 45982 stoweidlem48 46003 setrec1lem3 48919 setrec1 48921 |
Copyright terms: Public domain | W3C validator |