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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 4346 | . 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 1481 Vcvv 2689 suc csuc 4295 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-suc 4301 |
This theorem is referenced by: eqelsuc 4349 unon 4435 ordunisuc2r 4438 ordsoexmid 4485 limom 4535 0elnn 4540 tfrexlem 6239 tfri1dALT 6256 tfrcl 6269 frecabcl 6304 phplem4 6757 fiintim 6825 fidcenumlemr 6851 infnninf 7030 nnnninf 7031 prarloclemarch2 7251 prarloclemlt 7325 ennnfonelemex 11963 ennnfonelemrn 11968 bj-nn0suc0 13319 bj-nnelirr 13322 bj-inf2vnlem2 13340 bj-findis 13348 nninfsellemeq 13385 |
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