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Mirrors > Home > MPE Home > Th. List > brrelex1 | Structured version Visualization version GIF version |
Description: If two classes are related by a binary relation, then the first class is a set. (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
brrelex1 | ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brrelex12 5610 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | 1 | simpld 498 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 Vcvv 3415 class class class wbr 5062 Rel wrel 5565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pr 5331 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3417 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-sn 4551 df-pr 4553 df-op 4557 df-br 5063 df-opab 5125 df-xp 5566 df-rel 5567 |
This theorem is referenced by: brrelex1i 5614 posn 5643 frsn 5645 releldm 5822 relelrn 5823 relimasn 5961 funmo 6405 ertr 8415 dirtr 18121 eqvreltr 36470 frege129d 41063 nnfoctb 42283 clim2d 42904 climfv 42922 meadjiun 43694 caragenunicl 43752 |
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