Theorem epfrc 5436

Description: A subset of an epsilon-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 5416	. 2 ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
3		dfin5 3873	. . . . 5 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥}
4		epel 5364	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
5	4	rabbii 3421	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥}
6	3, 5	eqtr4i 2824	. . . 4 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥}
7	6	eqeq1i 2802	. . 3 ⊢ ((𝐵 ∩ 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
8	7	rexbii 3213	. 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅)
9	2, 8	sylibr 235	1 ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅)

Syntax hints:   → wi 4   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083   ≠ wne 2986  ∃wrex 3108  {crab 3111  Vcvv 3440   ∩ cin 3864   ⊆ wss 3865  ∅c0 4217   class class class wbr 4968   E cep 5359   Fr wfr 5406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-eprel 5360  df-fr 5409

This theorem is referenced by:  wefrc  5444  onfr  6112  epfrs  9026