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Mirrors > Home > MPE Home > Th. List > eluniab | Structured version Visualization version GIF version |
Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
eluniab | ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 4911 | . 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})) | |
2 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦 | |
3 | nfsab1 2717 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
4 | 2, 3 | nfan 1902 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) |
5 | nfv 1917 | . . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝜑) | |
6 | eleq2w 2817 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)) | |
7 | eleq1w 2816 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})) | |
8 | abid 2713 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
9 | 7, 8 | bitrdi 286 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) |
10 | 6, 9 | anbi12d 631 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝑥 ∧ 𝜑))) |
11 | 4, 5, 10 | cbvexv1 2338 | . 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑)) |
12 | 1, 11 | bitri 274 | 1 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 {cab 2709 ∪ cuni 4908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-uni 4909 |
This theorem is referenced by: elunirab 4924 dfiun2gOLD 5034 inuni 5343 elfv 6889 unielxp 8015 frrlem8 8280 frrlem10 8282 wfrlem12OLD 8322 tfrlem9 8387 dfac5lem2 10121 fin23lem30 10339 unisngl 23251 metrest 24253 aannenlem2 26066 fpwrelmapffslem 32212 dfiota3 35187 mptsnunlem 36522 nnoeomeqom 42364 |
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