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Theorem dfsn2ALT 4386
Description: Alternate definition of singleton, based on the (alternate) definition of unordered 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 929 . . 3 ((𝑥 = 𝐴𝑥 = 𝐴) ↔ 𝑥 = 𝐴)
21abbii 2914 . 2 {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐴)} = {𝑥𝑥 = 𝐴}
3 dfpr2 4385 . 2 {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐴)}
4 df-sn 4367 . 2 {𝐴} = {𝑥𝑥 = 𝐴}
52, 3, 43eqtr4ri 2830 1 {𝐴} = {𝐴, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  wo 874   = wceq 1653  {cab 2783  {csn 4366  {cpr 4368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385  df-un 3772  df-sn 4367  df-pr 4369
This theorem is referenced by: (None)
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