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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 3361 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V) | |
2 | elex 3361 | . . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ V) | |
3 | 2 | adantr 466 | . . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V) |
4 | 3 | exlimiv 2009 | . 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V) |
5 | eleq1 2837 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
6 | 5 | anbi1d 607 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
7 | 6 | exbidv 2001 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
8 | df-uni 4573 | . . 3 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)} | |
9 | 7, 8 | elab2g 3502 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
10 | 1, 4, 9 | pm5.21nii 367 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1630 ∃wex 1851 ∈ wcel 2144 Vcvv 3349 ∪ cuni 4572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-v 3351 df-uni 4573 |
This theorem is referenced by: eluni2 4576 elunii 4577 eluniab 4583 uniun 4591 uniin 4592 uniss 4593 unissb 4603 dftr2 4886 unipw 5046 dmuni 5472 fununi 6104 elunirn 6651 uniex2 7098 uniuni 7117 mpt2xopxnop0 7492 wfrfun 7577 wfrlem17 7583 tfrlem7 7631 tfrlem9a 7634 inf2 8683 inf3lem2 8689 rankwflemb 8819 cardprclem 9004 carduni 9006 iunfictbso 9136 kmlem3 9175 kmlem4 9176 cfub 9272 isf34lem4 9400 grothtsk 9858 suplem1pr 10075 toprntopon 20949 isbasis2g 20972 tgval2 20980 ntreq0 21101 cmpsublem 21422 cmpsub 21423 cmpcld 21425 is1stc2 21465 alexsubALTlem3 22072 alexsubALT 22074 frrlem5c 32117 fnessref 32683 bj-restuni 33375 truniALT 39270 truniALTVD 39630 unisnALT 39678 elunif 39691 ssfiunibd 40034 stoweidlem27 40755 stoweidlem48 40776 setrec1lem3 42954 setrec1 42956 |
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