Theorem dfsn2ALT 4616

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.

Step	Hyp	Ref	Expression
1		oridm 917	. . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐴) ↔ 𝑥 = 𝐴)
2	1	abbii 2836	. 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} = {𝑥 ∣ 𝑥 = 𝐴}
3		dfpr2 4615	. 2 ⊢ {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)}
4		df-sn 4595	. 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
5	2, 3, 4	3eqtr4ri 2803	1 ⊢ {𝐴} = {𝐴, 𝐴}

Syntax hints:   ∨ wo 860   = wceq 1567  {cab 2747  {csn 4594  {cpr 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-sn 4595  df-pr 4597

This theorem is referenced by: (None)