Theorem fr0 2179

Description: Any relation is founded on the empty set.

Assertion

Ref	Expression
fr0	⊢ R Fr ∅

Proof of Theorem fr0

Step	Hyp	Ref	Expression
1		dffr2 2171	. 2 ⊢ (R Fr ∅ ↔ ∀x((x ⊆ ∅ ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zRy}) = ∅))
2		ss0 1727	. . . 4 ⊢ (x ⊆ ∅ → x = ∅)
3		iman 205	. . . 4 ⊢ ((x ⊆ ∅ → x = ∅) ↔ ¬ (x ⊆ ∅ ∧ ¬ x = ∅))
4	2, 3	mpbi 164	. . 3 ⊢ ¬ (x ⊆ ∅ ∧ ¬ x = ∅)
5	4	pm2.21i 73	. 2 ⊢ ((x ⊆ ∅ ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zRy}) = ∅)
6	1, 5	mpgbir 686	1 ⊢ R Fr ∅

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  {cab 1090   = wceq 1091  ∃wrex 1202   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   Fr wfr 2061

This theorem is referenced by:  we0 2196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-fr 2169

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