Theorem selsALT 5408

Description: Alternate proof of sels 5407, requiring ax-sep 5248 but not using el 5405 (which is proved from it as elALT 5409). (especially when the proof of el 5405 is inlined in sels 5407). (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5409. (Revised by BJ, 3-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
selsALT	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem selsALT

Step	Hyp	Ref	Expression
1		snidg 4617	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		snexg 5397	. . 3 ⊢ (𝐴 ∈ {𝐴} → {𝐴} ∈ V)
3		snidg 4617	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ {𝐴})
4		eleq2 2849	. . 3 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
5	2, 3, 4	spcedv 3552	. 2 ⊢ (𝐴 ∈ {𝐴} → ∃𝑥 𝐴 ∈ 𝑥)
6	1, 5	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

This proof depends on syntax axioms:   → wi 4  ∃wex 1811   ∈ wcel 2144  Vcvv 3450  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 4151  df-sn 4585  df-pr 4587

This theorem is used by:  elALT  5409