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Theorem 0heALT 37580
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 37578 . 2 (∅ × 𝐴) hereditary 𝐴
2 0xp 5162 . . 3 (∅ × 𝐴) = ∅
3 heeq1 37574 . . 3 ((∅ × 𝐴) = ∅ → ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴))
42, 3ax-mp 5 . 2 ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴)
51, 4mpbi 220 1 ∅ hereditary 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  c0 3893   × cxp 5074   hereditary whe 37569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-xp 5082  df-rel 5083  df-cnv 5084  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-he 37570
This theorem is referenced by: (None)
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