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Mirrors > Home > MPE Home > Th. List > Mathboxes > eluni2f | Structured version Visualization version GIF version |
Description: Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
eluni2f.1 | ⊢ Ⅎ𝑥𝐴 |
eluni2f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
eluni2f | ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1861 | . 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
2 | eluni2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | eluni2f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | elunif 41280 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
5 | df-rex 3146 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
6 | 1, 4, 5 | 3bitr4i 305 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1780 ∈ wcel 2114 Ⅎwnfc 2963 ∃wrex 3141 ∪ cuni 4840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rex 3146 df-v 3498 df-uni 4841 |
This theorem is referenced by: smfresal 43070 smfpimbor1lem2 43081 |
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