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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 4055 | . . 3 | |
2 | strsl0.e | . . . 4 Slot | |
3 | 2 | simpli 110 | . . 3 Slot |
4 | 2 | simpri 112 | . . 3 |
5 | 1, 3, 4 | strnfvn 11980 | . 2 |
6 | 0fv 5456 | . 2 | |
7 | 5, 6 | eqtr2i 2161 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wcel 1480 c0 3363 cfv 5123 cn 8720 cnx 11956 Slot cslot 11958 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-slot 11963 |
This theorem is referenced by: base0 12008 |
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