Theorem dfsn2ALT 4595

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 2798	. 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} = {𝑥 ∣ 𝑥 = 𝐴}
3		dfpr2 4594	. 2 ⊢ {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)}
4		df-sn 4574	. 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
5	2, 3, 4	3eqtr4ri 2765	1 ⊢ {𝐴} = {𝐴, 𝐴}

Syntax hints:   ∨ wo 847   = wceq 1541  {cab 2709  {csn 4573  {cpr 4575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-sn 4574  df-pr 4576

This theorem is referenced by: (None)