Theorem elelsuc 6434

Description: Membership in a successor. (Contributed by NM, 20-Jun-1998.)

Assertion

Ref	Expression
elelsuc	⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)

Proof of Theorem elelsuc

Step	Hyp	Ref	Expression
1		orc 866	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
2		elsucg 6429	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
3	1, 2	mpbird 257	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1542   ∈ wcel 2107  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3952  df-sn 4628  df-suc 6367

This theorem is referenced by:  suctr  6447  pssnn  9164  pssnnOLD  9261  ttrcltr  9707  ttrclss  9711  ttrclselem2  9717  pwsdompw  10195  fin1a2lem4  10394  grur1a  10810  bnj570  33854  satom  34285  satfv0  34287  satfvsuc  34290  satf00  34303  satf0suc  34305  sat1el2xp  34308  fmla  34310  fmla0  34311  fmlasuc0  34313  satfdmfmla  34329  finxpsuclem  36216