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Theorem 0heALT 37097
 Description: The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
0heALT ∅ hereditary 𝐴

Proof of Theorem 0heALT
StepHypRef Expression
1 xphe 37095 . 2 (∅ × 𝐴) hereditary 𝐴
2 0xp 5122 . . 3 (∅ × 𝐴) = ∅
3 heeq1 37091 . . 3 ((∅ × 𝐴) = ∅ → ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴))
42, 3ax-mp 5 . 2 ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴)
51, 4mpbi 219 1 ∅ hereditary 𝐴
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ∅c0 3874   × cxp 5036   hereditary whe 37086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-he 37087 This theorem is referenced by: (None)
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