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Theorem dfsn2ALT 4645
Description: Alternate definition of singleton, based on the (alternate) definition of pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by AV, 12-Jun-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
dfsn2ALT {𝐴} = {𝐴, 𝐴}

Proof of Theorem dfsn2ALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oridm 903 . . 3 ((𝑥 = 𝐴𝑥 = 𝐴) ↔ 𝑥 = 𝐴)
21abbii 2798 . 2 {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐴)} = {𝑥𝑥 = 𝐴}
3 dfpr2 4644 . 2 {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐴)}
4 df-sn 4626 . 2 {𝐴} = {𝑥𝑥 = 𝐴}
52, 3, 43eqtr4ri 2767 1 {𝐴} = {𝐴, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  wo 846   = wceq 1534  {cab 2705  {csn 4625  {cpr 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-un 3950  df-sn 4626  df-pr 4628
This theorem is referenced by: (None)
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