Theorem idhe 44235

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 6009	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2		dfhe2 44222	. 2 ⊢ ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
3	1, 2	mpbir 231	1 ⊢ I hereditary 𝐴

Syntax hints:   ⊆ wss 3890   I cid 5519   × cxp 5623   ↾ cres 5627   hereditary whe 44220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-he 44221

This theorem is referenced by:  sshepw  44237