Theorem selsALT 5420

Description: Alternate proof of sels 5419, requiring ax-sep 5255 but not using el 5417 (which is proved from it as elALT 5421). (especially when the proof of el 5417 is inlined in sels 5419). (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5421. (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 4624	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		snexg 5409	. . 3 ⊢ (𝐴 ∈ {𝐴} → {𝐴} ∈ V)
3		snidg 4624	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ {𝐴})
4		eleq2 2851	. . 3 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
5	2, 3, 4	spcedv 3555	. 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 1812   ∈ wcel 2145  Vcvv 3453  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-sn 4588  df-pr 4590

This theorem is used by:  elALT  5421