Theorem ecidsn 8799

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 8746	. 2 ⊢ [𝐴] I = ( I “ {𝐴})
2		imai 6094	. 2 ⊢ ( I “ {𝐴}) = {𝐴}
3	1, 2	eqtri 2763	1 ⊢ [𝐴] I = {𝐴}

Syntax hints:   = wceq 1537  {csn 4631   I cid 5582   “ cima 5692  [cec 8742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746

This theorem is referenced by:  extid  38292