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Mirrors > Home > MPE Home > Th. List > abrexex | Structured version Visualization version GIF version |
Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7894. See also abrexex2 7903. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
abrexex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
abrexex | ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abrexex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | abrexexg 7894 | . 2 ⊢ (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {cab 2714 ∃wrex 3074 Vcvv 3446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-rep 5243 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-v 3448 |
This theorem is referenced by: ab2rexex 7913 kmlem10 10096 cshwsexa 14713 shftfval 14956 dvdsrval 20075 cmpsublem 22753 cmpsub 22754 ptrescn 22993 addsproplem2 27285 negsid 27342 satfvsuclem1 33956 fmlasuc0 33981 heibor1lem 36271 pointsetN 38207 eldiophb 41083 |
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