Theorem elsuc2g 6455

Description: Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)

Assertion

Ref	Expression
elsuc2g	⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))

Proof of Theorem elsuc2g

Step	Hyp	Ref	Expression
1		df-suc 6392	. . 3 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2	1	eleq2i 2831	. 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵}))
3		elun 4163	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
4		elsn2g 4669	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
5	4	orbi2d 915	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
6	3, 5	bitrid 283	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	2, 6	bitrid 283	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1537   ∈ wcel 2106   ∪ cun 3961  {csn 4631  suc csuc 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-suc 6392

This theorem is referenced by:  elsuc2  6457  om2uzlti  13988