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Mirrors > Home > ILE Home > Th. List > sucidg | GIF version |
Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
Ref | Expression |
---|---|
sucidg | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 732 | . 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴) |
3 | elsucg 4400 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 168 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 708 = wceq 1353 ∈ wcel 2148 suc csuc 4361 |
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 3597 df-suc 4367 |
This theorem is referenced by: sucid 4413 nsuceq0g 4414 trsuc 4418 sucssel 4420 ordsucg 4497 sucunielr 4505 suc11g 4552 nlimsucg 4561 peano2b 4610 omsinds 4617 nnpredlt 4619 frecsuclem 6400 phplem4dom 6855 phplem4on 6860 dif1en 6872 fin0 6878 fin0or 6879 fidcenumlemrks 6945 bj-peano4 14329 |
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