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Mirrors > Home > ILE Home > Th. List > ctex | GIF version |
Description: A class dominated by ω is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.) |
Ref | Expression |
---|---|
ctex | ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 6590 | . 2 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 4540 | 1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1461 Vcvv 2655 class class class wbr 3893 ωcom 4462 ≼ cdom 6584 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-br 3894 df-opab 3948 df-xp 4503 df-rel 4504 df-dom 6587 |
This theorem is referenced by: cnvct 6654 ssct 6662 xpct 11747 |
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