Theorem selsALT 5445

Description: Alternate proof of sels 5444, requiring ax-sep 5303 but not using el 5443 (which is proved from it as elALT 5446). (especially when the proof of el 5443 is inlined in sels 5444). (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5446. (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 4667	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		snexg 5436	. . 3 ⊢ (𝐴 ∈ {𝐴} → {𝐴} ∈ V)
3		snidg 4667	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ {𝐴})
4		eleq2 2818	. . 3 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
5	2, 3, 4	spcedv 3587	. 2 ⊢ (𝐴 ∈ {𝐴} → ∃𝑥 𝐴 ∈ 𝑥)
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4  ∃wex 1773   ∈ wcel 2098  Vcvv 3473  {csn 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-un 3954  df-sn 4633  df-pr 4635

This theorem is referenced by:  elALT  5446