Theorem epfrc 5640

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 5618	. 2 ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
3		dfin5 3907	. . . . 5 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥}
4		epel 5558	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
5	4	rabbii 3417	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥}
6	3, 5	eqtr4i 2786	. . . 4 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥}
7	6	eqeq1i 2765	. . 3 ⊢ ((𝐵 ∩ 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
8	7	rexbii 3109	. 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) = ∅)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  {crab 3412  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103   E cep 5554   Fr wfr 5605

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  ax-sep 5251  ax-pr 5398

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-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-eprel 5555  df-fr 5608

This theorem is used by:  wefrc  5649  onfr  6397  epfrs  9710