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| Mirrors > Home > ILE Home > Th. List > brrelex2i | GIF version | ||
| Description: The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
| Ref | Expression |
|---|---|
| brrelexi.1 | ⊢ Rel 𝑅 |
| Ref | Expression |
|---|---|
| brrelex2i | ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brrelexi.1 | . 2 ⊢ Rel 𝑅 | |
| 2 | brrelex2 4715 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) | |
| 3 | 1, 2 | mpan 424 | 1 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∈ wcel 2175 Vcvv 2771 class class class wbr 4043 Rel wrel 4679 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-br 4044 df-opab 4105 df-xp 4680 df-rel 4681 |
| This theorem is referenced by: vtoclr 4722 brdomi 6837 xpdom2 6925 xpdom1g 6927 mapdom1g 6943 djudom 7194 difinfsn 7201 enomnilem 7239 enmkvlem 7262 enwomnilem 7270 djuenun 7323 aprcl 8718 hashinfom 10921 clim 11563 ntrivcvgap0 11831 |
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