Theorem sucidg 4119

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.)

Assertion

Ref	Expression
sucidg	⊢ (A ∈ 𝑉 → A ∈ suc A)

Proof of Theorem sucidg

Step	Hyp	Ref	Expression
1		eqid 2037	. . 3 ⊢ A = A
2	1	olci 650	. 2 ⊢ (A ∈ A ∨ A = A)
3		elsucg 4107	. 2 ⊢ (A ∈ 𝑉 → (A ∈ suc A ↔ (A ∈ A ∨ A = A)))
4	2, 3	mpbiri 157	1 ⊢ (A ∈ 𝑉 → A ∈ suc A)

Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242   ∈ wcel 1390  suc csuc 4068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-suc 4074

This theorem is referenced by:  sucid  4120  nsuceq0g  4121  trsuc  4125  sucssel  4127  ordsucg  4194  sucunielr  4201  suc11g  4235  nlimsucg  4242  onpsssuc  4247  peano2b  4280  frecsuclem2  5928  bj-peano4  9343