Theorem 0heALT 44461

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

Step	Hyp	Ref	Expression
1		xphe 44459	. 2 ⊢ (∅ × 𝐴) hereditary 𝐴
2		0xp 5764	. . 3 ⊢ (∅ × 𝐴) = ∅
3		heeq1 44455	. . 3 ⊢ ((∅ × 𝐴) = ∅ → ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴)
5	1, 4	mpbi 233	1 ⊢ ∅ hereditary 𝐴

Syntax hints:   ↔ wb 209   = wceq 1568  ∅c0 4294   × cxp 5663   hereditary whe 44450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-11 2199  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-he 44451

This theorem is referenced by: (None)