Theorem 0heALT 43355

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 43353	. 2 ⊢ (∅ × 𝐴) hereditary 𝐴
2		0xp 5776	. . 3 ⊢ (∅ × 𝐴) = ∅
3		heeq1 43349	. . 3 ⊢ ((∅ × 𝐴) = ∅ → ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴)
5	1, 4	mpbi 229	1 ⊢ ∅ hereditary 𝐴

Syntax hints:   ↔ wb 205   = wceq 1533  ∅c0 4322   × cxp 5676   hereditary whe 43344

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 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-he 43345

This theorem is referenced by: (None)