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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 4116 | . . 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 12437 | . 2 ⊢ (𝐸‘∅) = (∅‘(𝐸‘ndx)) |
6 | 0fv 5531 | . 2 ⊢ (∅‘(𝐸‘ndx)) = ∅ | |
7 | 5, 6 | eqtr2i 2192 | 1 ⊢ ∅ = (𝐸‘∅) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∈ wcel 2141 ∅c0 3414 ‘cfv 5198 ℕcn 8878 ndxcnx 12413 Slot cslot 12415 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fv 5206 df-slot 12420 |
This theorem is referenced by: base0 12465 |
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