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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 7929. See also abrexex2 7938. (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 7929 | . 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 1541 ∈ wcel 2106 {cab 2708 ∃wrex 3069 Vcvv 3473 |
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-ext 2702 ax-rep 5278 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-rex 3070 df-v 3475 |
This theorem is referenced by: ab2rexex 7948 kmlem10 10136 cshwsexa 14756 shftfval 14999 dvdsrval 20127 cmpsublem 22832 cmpsub 22833 ptrescn 23072 addsproplem2 27370 negsid 27431 satfvsuclem1 34179 fmlasuc0 34204 heibor1lem 36480 pointsetN 38415 eldiophb 41264 |
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