Theorem ecref 8741

Description: All elements are in their own equivalence class. (Contributed by Thierry Arnoux, 14-Feb-2025.)

Assertion

Ref	Expression
ecref	⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ [𝐴]𝑅)

Proof of Theorem ecref

Step	Hyp	Ref	Expression
1		simpl 487	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝑅 Er 𝑋)
2		simpr 489	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3	1, 2	erref 8716	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
4		elecg 8740	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
5	2, 4	sylancom 599	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
6	3, 5	mpbird 260	1 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ [𝐴]𝑅)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2143   class class class wbr 5110   Er wer 8692  [cec 8693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-er 8695  df-ec 8697

This theorem is referenced by:  ghmqusnsglem1  19351  ghmqusnsglem2  19352  ghmquskerlem1  19354  ghmquskerlem2  19356  qsdrnglem2  33756