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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 5268 | . . 3 ⊢ ∅ ∈ V | |
2 | str0.a | . . 3 ⊢ 𝐹 = Slot 𝐼 | |
3 | 1, 2 | strfvn 17066 | . 2 ⊢ (𝐹‘∅) = (∅‘𝐼) |
4 | 0fv 6890 | . 2 ⊢ (∅‘𝐼) = ∅ | |
5 | 3, 4 | eqtr2i 2762 | 1 ⊢ ∅ = (𝐹‘∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∅c0 4286 ‘cfv 6500 Slot cslot 17061 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
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-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-slot 17062 |
This theorem is referenced by: strfvi 17070 setsnid 17089 setsnidOLD 17090 base0 17096 resseqnbas 17130 resslemOLD 17131 oppchomfval 17602 oppchomfvalOLD 17603 fuchom 17857 fuchomOLD 17858 xpchomfval 18075 xpccofval 18078 oduleval 18186 0pos 18218 0posOLD 18219 frmdplusg 18672 efmndplusg 18698 oppgplusfval 19134 mgpplusg 19908 opprmulfval 20059 sralem 20683 sralemOLD 20684 srasca 20691 srascaOLD 20692 sravsca 20693 sravscaOLD 20694 sraip 20695 zlmlem 20940 zlmlemOLD 20941 zlmvsca 20949 thlle 21125 thlleOLD 21126 thloc 21128 psrplusg 21372 psrmulr 21375 psrvscafval 21381 opsrle 21471 ply1plusgfvi 21636 psr1sca2 21645 ply1sca2 21648 resstopn 22560 tnglem 24019 tnglemOLD 24020 tngds 24034 tngdsOLD 24035 ttglem 27868 ttglemOLD 27869 iedgval0 28040 resvlem 32176 resvlemOLD 32177 mendplusgfval 41559 mendmulrfval 41561 mendsca 41563 mendvscafval 41564 |
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