Theorem eqsucidi 35787

Description: A set belongs to its successor. Inference form of sucidg 6461 with an equality hypothesis. (Contributed by BTernaryTau, 19-Oct-2026.)

Hypotheses

Ref	Expression
eqsucidi.1	⊢ 𝐴 ∈ 𝑉
eqsucidi.2	⊢ 𝐵 = suc 𝐴

Assertion

Ref	Expression
eqsucidi	⊢ 𝐴 ∈ 𝐵

Proof of Theorem eqsucidi

Step	Hyp	Ref	Expression
1		eqsucidi.1	. . 3 ⊢ 𝐴 ∈ 𝑉
2		sucidg 6461	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	. 2 ⊢ 𝐴 ∈ suc 𝐴
4		eqsucidi.2	. 2 ⊢ 𝐵 = suc 𝐴
5	3, 4	eleqtrri 2863	1 ⊢ 𝐴 ∈ 𝐵

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  suc csuc 6379

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-un 4154  df-sn 4589  df-suc 6383

This theorem is used by:  gdlopcfv2  35972  gdlopcfv3  35973  gdlopcfv4  35974  gdlopcfv5  35975  gdlopcfv6  35976  gdlopcfv7  35977  gdlopcfv8  35978