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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0he | Structured version Visualization version GIF version |
Description: The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) |
Ref | Expression |
---|---|
0he | ⊢ ∅ hereditary 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ima 6029 | . . 3 ⊢ (∅ “ 𝐴) = ∅ | |
2 | 0ss 4355 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
3 | 1, 2 | eqsstri 3977 | . 2 ⊢ (∅ “ 𝐴) ⊆ 𝐴 |
4 | df-he 41950 | . 2 ⊢ (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴) | |
5 | 3, 4 | mpbir 230 | 1 ⊢ ∅ hereditary 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3909 ∅c0 4281 “ cima 5635 hereditary whe 41949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-xp 5638 df-cnv 5640 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-he 41950 |
This theorem is referenced by: (None) |
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