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Mirrors > Home > MPE Home > Th. List > fi0 | Structured version Visualization version GIF version |
Description: The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
fi0 | ⊢ (fi‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4790 | . . 3 ⊢ ∅ ∈ V | |
2 | fival 8318 | . . 3 ⊢ (∅ ∈ V → (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥}) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥} |
4 | vprc 4796 | . . . 4 ⊢ ¬ V ∈ V | |
5 | id 22 | . . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 → 𝑦 = ∩ 𝑥) | |
6 | inss1 3833 | . . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) ⊆ 𝒫 ∅ | |
7 | 6 | sseli 3599 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅) |
8 | elpwi 4168 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅) | |
9 | ss0 3974 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅) |
11 | 10 | inteqd 4480 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = ∩ ∅) |
12 | int0 4490 | . . . . . . . 8 ⊢ ∩ ∅ = V | |
13 | 11, 12 | syl6eq 2672 | . . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = V) |
14 | 5, 13 | sylan9eqr 2678 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = V) |
15 | 14 | rexlimiva 3028 | . . . . 5 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 = V) |
16 | vex 3203 | . . . . 5 ⊢ 𝑦 ∈ V | |
17 | 15, 16 | syl6eqelr 2710 | . . . 4 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → V ∈ V) |
18 | 4, 17 | mto 188 | . . 3 ⊢ ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 |
19 | 18 | abf 3978 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥} = ∅ |
20 | 3, 19 | eqtri 2644 | 1 ⊢ (fi‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 ∩ cint 4475 ‘cfv 5888 Fincfn 7955 ficfi 8316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-fi 8317 |
This theorem is referenced by: fieq0 8327 firest 16093 restbas 20962 |
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