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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 5728 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | 1 | simpld 494 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 Vcvv 3473 class class class wbr 5148 Rel wrel 5681 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 |
This theorem is referenced by: brrelex1i 5732 posn 5761 frsn 5763 releldm 5943 relelrn 5944 relimasn 6083 funmo 6563 funmoOLD 6564 ertr 8721 dirtr 18560 eqvreltr 37781 fsuppss 41372 frege129d 42817 nnfoctb 44036 clim2d 44688 climfv 44706 meadjiun 45481 caragenunicl 45539 |
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