Theorem ercl2 8758

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 8757	. 2 ⊢ (𝜑 → 𝐵𝑅𝐴)
4	1, 3	ercl 8756	1 ⊢ (𝜑 → 𝐵 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2108   class class class wbr 5143   Er wer 8742

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-er 8745

This theorem is referenced by:  qliftfun  8842  efgcpbl2  19775  frgpcpbl  19777  prjspner1  42636