Theorem idhe 43751

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 6080	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2		dfhe2 43738	. 2 ⊢ ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
3	1, 2	mpbir 231	1 ⊢ I hereditary 𝐴

Syntax hints:   ⊆ wss 3976   I cid 5592   × cxp 5698   ↾ cres 5702   hereditary whe 43736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-he 43737

This theorem is referenced by:  sshepw  43753