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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 45021 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) | 
| 5 | df-rex 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
| 6 | 1, 4, 5 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 Ⅎwnfc 2890 ∃wrex 3070 ∪ cuni 4907 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 df-v 3482 df-uni 4908 | 
| This theorem is referenced by: smfresal 46803 smfpimbor1lem2 46814 | 
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