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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 4416 | . 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 2148 Vcvv 2737 suc csuc 4365 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3598 df-suc 4371 |
This theorem is referenced by: eqelsuc 4419 unon 4510 ordunisuc2r 4513 ordsoexmid 4561 limom 4613 0elnn 4618 tfrexlem 6334 tfri1dALT 6351 tfrcl 6364 frecabcl 6399 phplem4 6854 fiintim 6927 fidcenumlemr 6953 nninfwlpoimlemginf 7173 pw1ne3 7228 sucpw1ne3 7230 sucpw1nel3 7231 prarloclemarch2 7417 prarloclemlt 7491 ennnfonelemex 12414 ennnfonelemrn 12419 bj-nn0suc0 14672 bj-nnelirr 14675 bj-inf2vnlem2 14693 bj-findis 14701 nninfsellemeq 14733 |
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