Theorem ecref 8680

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 8655	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
4		elecg 8679	. . 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 2113   class class class wbr 5098   Er wer 8632  [cec 8633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-er 8635  df-ec 8637

This theorem is referenced by:  ghmqusnsglem1  19209  ghmqusnsglem2  19210  ghmquskerlem1  19212  ghmquskerlem2  19214  qsdrnglem2  33577