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Mirrors > Home > ILE Home > Th. List > strsl0 | GIF 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 | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
Ref | Expression |
---|---|
strsl0 | ⊢ ∅ = (𝐸‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4063 | . . 3 ⊢ ∅ ∈ V | |
2 | strsl0.e | . . . 4 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | |
3 | 2 | simpli 110 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 2 | simpri 112 | . . 3 ⊢ (𝐸‘ndx) ∈ ℕ |
5 | 1, 3, 4 | strnfvn 12019 | . 2 ⊢ (𝐸‘∅) = (∅‘(𝐸‘ndx)) |
6 | 0fv 5464 | . 2 ⊢ (∅‘(𝐸‘ndx)) = ∅ | |
7 | 5, 6 | eqtr2i 2162 | 1 ⊢ ∅ = (𝐸‘∅) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∈ wcel 1481 ∅c0 3368 ‘cfv 5131 ℕcn 8744 ndxcnx 11995 Slot cslot 11997 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fv 5139 df-slot 12002 |
This theorem is referenced by: base0 12047 |
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