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Mirrors > Home > MPE Home > Th. List > Mathboxes > he0 | Structured version Visualization version GIF version |
Description: Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.) |
Ref | Expression |
---|---|
he0 | ⊢ 𝐴 hereditary ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ima0 5726 | . . 3 ⊢ (𝐴 “ ∅) = ∅ | |
2 | 1 | eqimssi 3884 | . 2 ⊢ (𝐴 “ ∅) ⊆ ∅ |
3 | df-he 38902 | . 2 ⊢ (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅) | |
4 | 2, 3 | mpbir 223 | 1 ⊢ 𝐴 hereditary ∅ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3798 ∅c0 4146 “ cima 5349 hereditary whe 38901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-cnv 5354 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-he 38902 |
This theorem is referenced by: (None) |
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