Theorem ecidsn 8818

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 8765	. 2 ⊢ [𝐴] I = ( I “ {𝐴})
2		imai 6103	. 2 ⊢ ( I “ {𝐴}) = {𝐴}
3	1, 2	eqtri 2768	1 ⊢ [𝐴] I = {𝐴}

Syntax hints:   = wceq 1537  {csn 4648   I cid 5592   “ cima 5703  [cec 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765

This theorem is referenced by:  extid  38266