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Mirrors > Home > ILE Home > Th. List > strsl0 | Unicode version |
Description: All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.) |
Ref | Expression |
---|---|
strsl0.e | Slot |
Ref | Expression |
---|---|
strsl0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4103 | . . 3 | |
2 | strsl0.e | . . . 4 Slot | |
3 | 2 | simpli 110 | . . 3 Slot |
4 | 2 | simpri 112 | . . 3 |
5 | 1, 3, 4 | strnfvn 12352 | . 2 |
6 | 0fv 5515 | . 2 | |
7 | 5, 6 | eqtr2i 2186 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wcel 2135 c0 3404 cfv 5182 cn 8848 cnx 12328 Slot cslot 12330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fv 5190 df-slot 12335 |
This theorem is referenced by: base0 12380 |
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