Theorem ecref 8791

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 482	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝑅 Er 𝑋)
2		simpr 484	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3	1, 2	erref 8766	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
4		elecg 8790	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
5	2, 4	sylancom 588	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
6	3, 5	mpbird 257	1 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ [𝐴]𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2107   class class class wbr 5142   Er wer 8743  [cec 8744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-er 8746  df-ec 8748

This theorem is referenced by:  ghmqusnsglem1  19299  ghmqusnsglem2  19300  ghmquskerlem1  19302  ghmquskerlem2  19304  qsdrnglem2  33525