Theorem dfsn2ALT 4649

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 904	. . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐴) ↔ 𝑥 = 𝐴)
2	1	abbii 2803	. 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} = {𝑥 ∣ 𝑥 = 𝐴}
3		dfpr2 4648	. 2 ⊢ {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)}
4		df-sn 4630	. 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
5	2, 3, 4	3eqtr4ri 2772	1 ⊢ {𝐴} = {𝐴, 𝐴}

Syntax hints:   ∨ wo 846   = wceq 1542  {cab 2710  {csn 4629  {cpr 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-sn 4630  df-pr 4632

This theorem is referenced by: (None)