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Theorem selsALT 5432
Description: Alternate proof of sels 5431, requiring ax-sep 5292 but not using el 5430 (which is proved from it as elALT 5433). (especially when the proof of el 5430 is inlined in sels 5431). (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5433. (Revised by BJ, 3-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
selsALT (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem selsALT
StepHypRef Expression
1 snidg 4657 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 snexg 5423 . . 3 (𝐴 ∈ {𝐴} → {𝐴} ∈ V)
3 snidg 4657 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ {𝐴})
4 eleq2 2816 . . 3 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
52, 3, 4spcedv 3582 . 2 (𝐴 ∈ {𝐴} → ∃𝑥 𝐴𝑥)
61, 5syl 17 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1773  wcel 2098  Vcvv 3468  {csn 4623
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 2697  ax-sep 5292  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948  df-sn 4624  df-pr 4626
This theorem is referenced by:  elALT  5433
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