Theorem ecidsn 8729

Description: An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)

Assertion

Ref	Expression
ecidsn	⊢ [𝐴] I = {𝐴}

Proof of Theorem ecidsn

Step	Hyp	Ref	Expression
1		df-ec 8673	. 2 ⊢ [𝐴] I = ( I “ {𝐴})
2		imai 6045	. 2 ⊢ ( I “ {𝐴}) = {𝐴}
3	1, 2	eqtri 2752	1 ⊢ [𝐴] I = {𝐴}

Syntax hints:   = wceq 1540  {csn 4589   I cid 5532   “ cima 5641  [cec 8669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673

This theorem is referenced by:  extid  38298