Theorem prtlem9 39841

Description: Lemma for prter3 39859. (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 3237	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
2		eceq1 8735	. . 3 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
3	2	reximi 3100	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )
4	1, 3	sylbi 220	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  [cec 8693

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8697

This theorem is used by: (None)