Theorem selsALT 5424

Description: Alternate proof of sels 5423, requiring ax-sep 5258 but not using el 5421 (which is proved from it as elALT 5425). (especially when the proof of el 5421 is inlined in sels 5423). (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5425. (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 4627	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		snexg 5413	. . 3 ⊢ (𝐴 ∈ {𝐴} → {𝐴} ∈ V)
3		snidg 4627	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ {𝐴})
4		eleq2 2852	. . 3 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
5	2, 3, 4	spcedv 3558	. 2 ⊢ (𝐴 ∈ {𝐴} → ∃𝑥 𝐴 ∈ 𝑥)
6	1, 5	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  {csn 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3911  df-sn 4591  df-pr 4593

This theorem is referenced by:  elALT  5425