Theorem epfrc 5537

Description: A subset of a well-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.)

Hypothesis

Ref	Expression
epfrc.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
epfrc	⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem epfrc

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		epfrc.1	. . 3 ⊢ 𝐵 ∈ V
2	1	frc 5517	. 2 ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
3		dfin5 3874	. . . . 5 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥}
4		epel 5463	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
5	4	rabbii 3383	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥}
6	3, 5	eqtr4i 2768	. . . 4 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥}
7	6	eqeq1i 2742	. . 3 ⊢ ((𝐵 ∩ 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
8	7	rexbii 3170	. 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
9	2, 8	sylibr 237	1 ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅)

Syntax hints:   → wi 4   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∃wrex 3062  {crab 3065  Vcvv 3408   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237   class class class wbr 5053   E cep 5459   Fr wfr 5506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-eprel 5460  df-fr 5509

This theorem is referenced by:  wefrc  5545  onfr  6252  epfrs  9347