Theorem ecref 8686

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 483	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝑅 Er 𝑋)
2		simpr 485	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3	1, 2	erref 8661	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
4		elecg 8685	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
5	2, 4	sylancom 594	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
6	3, 5	mpbird 258	1 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ [𝐴]𝑅)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119   class class class wbr 5079   Er wer 8637  [cec 8638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-er 8640  df-ec 8642

This theorem is referenced by:  ghmqusnsglem1  19253  ghmqusnsglem2  19254  ghmquskerlem1  19256  ghmquskerlem2  19258  qsdrnglem2  33586