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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 6664 | . 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 2114 ∃!weu 2653 Vcvv 3496 class class class wbr 5068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 ax-pow 5268 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 |
This theorem is referenced by: eubrdm 43278 afv2eu 43444 |
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