Theorem ecref 8689

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 8664	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
4		elecg 8688	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
5	2, 4	sylancom 589	. 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 2114   class class class wbr 5085   Er wer 8640  [cec 8641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-er 8643  df-ec 8645

This theorem is referenced by:  ghmqusnsglem1  19255  ghmqusnsglem2  19256  ghmquskerlem1  19258  ghmquskerlem2  19260  qsdrnglem2  33556