Theorem prtlem9 38857

Description: Lemma for prter3 38875. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Assertion

Ref	Expression
prtlem9	⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   ∼ (𝑥)

Proof of Theorem prtlem9

Step	Hyp	Ref	Expression
1		risset 3212	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
2		eceq1 8710	. . 3 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
3	2	reximi 3067	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )
4	1, 3	sylbi 217	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  [cec 8669

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

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-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673

This theorem is referenced by: (None)