Theorem prtlem9 36878

Description: Lemma for prter3 36896. (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 3194	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
2		eceq1 8536	. . 3 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
3	2	reximi 3178	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )
4	1, 3	sylbi 216	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  [cec 8496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500

This theorem is referenced by: (None)