Theorem epfrc 5616

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 5594	. 2 ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
3		dfin5 3897	. . . . 5 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥}
4		epel 5534	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
5	4	rabbii 3394	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥}
6	3, 5	eqtr4i 2762	. . . 4 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥}
7	6	eqeq1i 2741	. . 3 ⊢ ((𝐵 ∩ 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
8	7	rexbii 3084	. 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
9	2, 8	sylibr 234	1 ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  {crab 3389  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273   class class class wbr 5085   E cep 5530   Fr wfr 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-eprel 5531  df-fr 5584

This theorem is referenced by:  wefrc  5625  onfr  6362  epfrs  9652