Theorem ecidsn 8807

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 8750	. 2 ⊢ [𝐴] I = ( I “ {𝐴})
2		imai 6085	. 2 ⊢ ( I “ {𝐴}) = {𝐴}
3	1, 2	eqtri 2786	1 ⊢ [𝐴] I = {𝐴}

This proof depends on syntax axioms:   = wceq 1570  {csn 4588   I cid 5550   “ cima 5659  [cec 8746

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ec 8750

This theorem is used by:  extid  39451