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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 4401 | . 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 2141 Vcvv 2730 suc csuc 4350 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3589 df-suc 4356 |
This theorem is referenced by: eqelsuc 4404 unon 4495 ordunisuc2r 4498 ordsoexmid 4546 limom 4598 0elnn 4603 tfrexlem 6313 tfri1dALT 6330 tfrcl 6343 frecabcl 6378 phplem4 6833 fiintim 6906 fidcenumlemr 6932 nninfwlpoimlemginf 7152 pw1ne3 7207 sucpw1ne3 7209 sucpw1nel3 7210 prarloclemarch2 7381 prarloclemlt 7455 ennnfonelemex 12369 ennnfonelemrn 12374 bj-nn0suc0 13985 bj-nnelirr 13988 bj-inf2vnlem2 14006 bj-findis 14014 nninfsellemeq 14047 |
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