Theorem idhe 44038

Description: The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)

Assertion

Ref	Expression
idhe	⊢ I hereditary 𝐴

Proof of Theorem idhe

Step	Hyp	Ref	Expression
1		idssxp 6008	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2		dfhe2 44025	. 2 ⊢ ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
3	1, 2	mpbir 231	1 ⊢ I hereditary 𝐴

Syntax hints:   ⊆ wss 3901   I cid 5518   × cxp 5622   ↾ cres 5626   hereditary whe 44023

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 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-he 44024

This theorem is referenced by:  sshepw  44040