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Theorem 0heALT 43216
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 43214 . 2 (∅ × 𝐴) hereditary 𝐴
2 0xp 5778 . . 3 (∅ × 𝐴) = ∅
3 heeq1 43210 . . 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 4324   × cxp 5678   hereditary whe 43205
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-he 43206
This theorem is referenced by: (None)
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