Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 5939 | . . 3 ⊢ (𝐴 “ ∅) = ∅ | |
2 | 1 | eqimssi 4024 | . 2 ⊢ (𝐴 “ ∅) ⊆ ∅ |
3 | df-he 40112 | . 2 ⊢ (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅) | |
4 | 2, 3 | mpbir 233 | 1 ⊢ 𝐴 hereditary ∅ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3935 ∅c0 4290 “ cima 5552 hereditary whe 40111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-he 40112 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |