Theorem 0heALT 44139

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 44137	. 2 ⊢ (∅ × 𝐴) hereditary 𝐴
2		0xp 5731	. . 3 ⊢ (∅ × 𝐴) = ∅
3		heeq1 44133	. . 3 ⊢ ((∅ × 𝐴) = ∅ → ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴)
5	1, 4	mpbi 230	1 ⊢ ∅ hereditary 𝐴

Syntax hints:   ↔ wb 206   = wceq 1542  ∅c0 4287   × cxp 5630   hereditary whe 44128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-he 44129

This theorem is referenced by: (None)