Theorem 0heALT 44360

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 44358	. 2 ⊢ (∅ × 𝐴) hereditary 𝐴
2		0xp 5747	. . 3 ⊢ (∅ × 𝐴) = ∅
3		heeq1 44354	. . 3 ⊢ ((∅ × 𝐴) = ∅ → ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴)
5	1, 4	mpbi 232	1 ⊢ ∅ hereditary 𝐴

Syntax hints:   ↔ wb 208   = wceq 1561  ∅c0 4286   × cxp 5646   hereditary whe 44349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-rel 5655  df-cnv 5656  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-he 44350

This theorem is referenced by: (None)