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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 7984. See also abrexex2 7993. (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 7984 | . 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 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 Vcvv 3478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-rep 5285 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-rex 3069 df-v 3480 |
This theorem is referenced by: ab2rexex 8003 kmlem10 10198 cshwsexa 14859 shftfval 15106 dvdsrval 20378 cmpsublem 23423 cmpsub 23424 ptrescn 23663 addsproplem2 28018 negsid 28088 onaddscl 28301 recut 28443 0reno 28444 satfvsuclem1 35344 fmlasuc0 35369 heibor1lem 37796 pointsetN 39724 eldiophb 42745 |
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