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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 5203 | . . 3 ⊢ ∅ ∈ V | |
2 | fival 8870 | . . 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 5211 | . . . 4 ⊢ ¬ V ∈ V | |
5 | id 22 | . . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 → 𝑦 = ∩ 𝑥) | |
6 | elinel1 4171 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅) | |
7 | elpwi 4550 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅) | |
8 | ss0 4351 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
9 | 6, 7, 8 | 3syl 18 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅) |
10 | 9 | inteqd 4873 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = ∩ ∅) |
11 | int0 4882 | . . . . . . . 8 ⊢ ∩ ∅ = V | |
12 | 10, 11 | syl6eq 2872 | . . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = V) |
13 | 5, 12 | sylan9eqr 2878 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = V) |
14 | 13 | rexlimiva 3281 | . . . . 5 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 = V) |
15 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
16 | 14, 15 | eqeltrrdi 2922 | . . . 4 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → V ∈ V) |
17 | 4, 16 | mto 199 | . . 3 ⊢ ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 |
18 | 17 | abf 4355 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥} = ∅ |
19 | 3, 18 | eqtri 2844 | 1 ⊢ (fi‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ∩ cint 4868 ‘cfv 6349 Fincfn 8503 ficfi 8868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-fi 8869 |
This theorem is referenced by: fieq0 8879 firest 16700 restbas 21760 |
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