Theorem ecidsn 8732

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 8676	. 2 ⊢ [𝐴] I = ( I “ {𝐴})
2		imai 6048	. 2 ⊢ ( I “ {𝐴}) = {𝐴}
3	1, 2	eqtri 2753	1 ⊢ [𝐴] I = {𝐴}

Syntax hints:   = wceq 1540  {csn 4592   I cid 5535   “ cima 5644  [cec 8672

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ec 8676

This theorem is referenced by:  extid  38305