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Theorem 0heALT 43091
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 43089 . 2 (∅ × 𝐴) hereditary 𝐴
2 0xp 5767 . . 3 (∅ × 𝐴) = ∅
3 heeq1 43085 . . 3 ((∅ × 𝐴) = ∅ → ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴))
42, 3ax-mp 5 . 2 ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴)
51, 4mpbi 229 1 ∅ hereditary 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  c0 4317   × cxp 5667   hereditary whe 43080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-he 43081
This theorem is referenced by: (None)
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