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Mirrors > Home > MPE Home > Th. List > str0 | Structured version Visualization version GIF version |
Description: All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
str0.a | ⊢ 𝐹 = Slot 𝐼 |
Ref | Expression |
---|---|
str0 | ⊢ ∅ = (𝐹‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5202 | . . 3 ⊢ ∅ ∈ V | |
2 | str0.a | . . 3 ⊢ 𝐹 = Slot 𝐼 | |
3 | 1, 2 | strfvn 16493 | . 2 ⊢ (𝐹‘∅) = (∅‘𝐼) |
4 | 0fv 6702 | . 2 ⊢ (∅‘𝐼) = ∅ | |
5 | 3, 4 | eqtr2i 2842 | 1 ⊢ ∅ = (𝐹‘∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∅c0 4288 ‘cfv 6348 Slot cslot 16470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-slot 16475 |
This theorem is referenced by: base0 16524 strfvi 16525 setsnid 16527 resslem 16545 oppchomfval 16972 fuchom 17219 xpchomfval 17417 xpccofval 17420 0pos 17552 oduleval 17729 frmdplusg 18007 oppgplusfval 18414 symgplusg 18445 mgpplusg 19172 opprmulfval 19304 sralem 19878 srasca 19882 sravsca 19883 sraip 19884 psrplusg 20089 psrmulr 20092 psrvscafval 20098 opsrle 20184 ply1plusgfvi 20338 psr1sca2 20347 ply1sca2 20350 zlmlem 20592 zlmvsca 20597 thlle 20769 thloc 20771 resstopn 21722 tnglem 23176 tngds 23184 ttglem 26589 iedgval0 26752 resvlem 30831 mendplusgfval 39663 mendmulrfval 39665 mendsca 39667 mendvscafval 39668 efmndplusg 43978 |
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