Theorem selsALT 5443

Description: Alternate proof of sels 5442, requiring ax-sep 5295 but not using el 5441 (which is proved from it as elALT 5444). (especially when the proof of el 5441 is inlined in sels 5442). (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5444. (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 4659	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		snexg 5434	. . 3 ⊢ (𝐴 ∈ {𝐴} → {𝐴} ∈ V)
3		snidg 4659	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ {𝐴})
4		eleq2 2829	. . 3 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
5	2, 3, 4	spcedv 3597	. 2 ⊢ (𝐴 ∈ {𝐴} → ∃𝑥 𝐴 ∈ 𝑥)
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4  ∃wex 1778   ∈ wcel 2107  Vcvv 3479  {csn 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-sn 4626  df-pr 4628

This theorem is referenced by:  elALT  5444