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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eubrv | Structured version Visualization version GIF version |
Description: If there is a unique set which is related to a class, then the class must be a set. (Contributed by AV, 25-Aug-2022.) |
Ref | Expression |
---|---|
eubrv | ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brprcneu 6438 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑏 𝐴𝑅𝑏) | |
2 | 1 | con4i 114 | 1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∃!weu 2585 Vcvv 3397 class class class wbr 4886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-nul 5025 ax-pow 5077 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4887 |
This theorem is referenced by: eubrdm 42093 afv2eu 42272 |
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