Theorem 0he 43022

Description: The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)

Assertion

Ref	Expression
0he	⊢ ∅ hereditary 𝐴

Proof of Theorem 0he

Step	Hyp	Ref	Expression
1		0ima 6067	. . 3 ⊢ (∅ “ 𝐴) = ∅
2		0ss 4388	. . 3 ⊢ ∅ ⊆ 𝐴
3	1, 2	eqsstri 4008	. 2 ⊢ (∅ “ 𝐴) ⊆ 𝐴
4		df-he 43013	. 2 ⊢ (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴)
5	3, 4	mpbir 230	1 ⊢ ∅ hereditary 𝐴

Syntax hints:   ⊆ wss 3940  ∅c0 4314   “ cima 5669   hereditary whe 43012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-he 43013

This theorem is referenced by: (None)