Theorem 0he 43885

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 6026	. . 3 ⊢ (∅ “ 𝐴) = ∅
2		0ss 4347	. . 3 ⊢ ∅ ⊆ 𝐴
3	1, 2	eqsstri 3976	. 2 ⊢ (∅ “ 𝐴) ⊆ 𝐴
4		df-he 43876	. 2 ⊢ (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴)
5	3, 4	mpbir 231	1 ⊢ ∅ hereditary 𝐴

Syntax hints:   ⊆ wss 3897  ∅c0 4280   “ cima 5617   hereditary whe 43875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-he 43876

This theorem is referenced by: (None)