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Theorem idhe 40862
 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
StepHypRef Expression
1 idssxp 5889 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2 dfhe2 40849 . 2 ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
31, 2mpbir 234 1 I hereditary 𝐴
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3859   I cid 5430   × cxp 5523   ↾ cres 5527   hereditary whe 40847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-he 40848 This theorem is referenced by:  sshepw  40864
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