Theorem elsuc2g 4456

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 4422	. . 3 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2	1	eleq2i 2273	. 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵}))
3		elun 3315	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
4		elsn2g 3667	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
5	4	orbi2d 792	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
6	3, 5	bitrid 192	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	2, 6	bitrid 192	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 710   = wceq 1373   ∈ wcel 2177   ∪ cun 3165  {csn 3634  suc csuc 4416

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3171  df-sn 3640  df-suc 4422

This theorem is referenced by:  elsuc2  4458  nntri3or  6586  frec2uzltd  10555