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Mirrors > Home > ILE Home > Th. List > sucid | GIF version |
Description: A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
Ref | Expression |
---|---|
sucid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
sucid | ⊢ 𝐴 ∈ suc 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucid.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sucidg 4333 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ suc 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2681 suc csuc 4282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-sn 3528 df-suc 4288 |
This theorem is referenced by: eqelsuc 4336 unon 4422 ordunisuc2r 4425 ordsoexmid 4472 limom 4522 0elnn 4527 tfrexlem 6224 tfri1dALT 6241 tfrcl 6254 frecabcl 6289 phplem4 6742 fiintim 6810 fidcenumlemr 6836 infnninf 7015 nnnninf 7016 prarloclemarch2 7220 prarloclemlt 7294 ennnfonelemex 11916 ennnfonelemrn 11921 bj-nn0suc0 13137 bj-nnelirr 13140 bj-inf2vnlem2 13158 bj-findis 13166 nninfsellemeq 13199 |
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