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Mirrors > Home > MPE Home > Th. List > rmorabex | Structured version Visualization version GIF version |
Description: Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.) |
Ref | Expression |
---|---|
rmorabex | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moabex 5343 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V) | |
2 | df-rmo 3146 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rab 3147 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
4 | 3 | eleq1i 2903 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V) |
5 | 1, 2, 4 | 3imtr4i 294 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∃*wmo 2616 {cab 2799 ∃*wrmo 3141 {crab 3142 Vcvv 3494 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rmo 3146 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4561 df-pr 4563 |
This theorem is referenced by: supexd 8911 |
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