Theorem ercl2 8638

Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ersym.1	⊢ (𝜑 → 𝑅 Er 𝑋)
ersym.2	⊢ (𝜑 → 𝐴𝑅𝐵)

Assertion

Ref	Expression
ercl2	⊢ (𝜑 → 𝐵 ∈ 𝑋)

Proof of Theorem ercl2

Step	Hyp	Ref	Expression
1		ersym.1	. 2 ⊢ (𝜑 → 𝑅 Er 𝑋)
2		ersym.2	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
3	1, 2	ersym 8637	. 2 ⊢ (𝜑 → 𝐵𝑅𝐴)
4	1, 3	ercl 8636	1 ⊢ (𝜑 → 𝐵 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2109   class class class wbr 5092   Er wer 8622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-er 8625

This theorem is referenced by:  qliftfun  8729  efgcpbl2  19636  frgpcpbl  19638  prjspner1  42599