![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eluni | 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 2771 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V) | |
2 | elex 2771 | . . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ V) | |
3 | 2 | adantr 276 | . . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V) |
4 | 3 | exlimiv 1609 | . 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V) |
5 | eleq1 2256 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
6 | 5 | anbi1d 465 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
7 | 6 | exbidv 1836 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
8 | df-uni 3836 | . . 3 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)} | |
9 | 7, 8 | elab2g 2907 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
10 | 1, 4, 9 | pm5.21nii 705 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2164 Vcvv 2760 ∪ cuni 3835 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-uni 3836 |
This theorem is referenced by: eluni2 3839 elunii 3840 eluniab 3847 uniun 3854 uniin 3855 uniss 3856 unissb 3865 dftr2 4129 unidif0 4196 unipw 4246 uniex2 4467 uniuni 4482 limom 4646 dmuni 4872 fununi 5322 nfvres 5588 elunirn 5809 tfrlem7 6370 tfrexlem 6387 tfrcldm 6416 fiuni 7037 isbasis2g 14213 tgval2 14219 ntreq0 14300 bj-uniex2 15408 |
Copyright terms: Public domain | W3C validator |