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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 5308 | . . 3 ⊢ ∅ ∈ V | |
2 | fival 9407 | . . 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 5316 | . . . 4 ⊢ ¬ V ∈ V | |
5 | id 22 | . . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 → 𝑦 = ∩ 𝑥) | |
6 | elinel1 4196 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅) | |
7 | elpwi 4610 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅) | |
8 | ss0 4399 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
9 | 6, 7, 8 | 3syl 18 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅) |
10 | 9 | inteqd 4956 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = ∩ ∅) |
11 | int0 4967 | . . . . . . . 8 ⊢ ∩ ∅ = V | |
12 | 10, 11 | eqtrdi 2789 | . . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = V) |
13 | 5, 12 | sylan9eqr 2795 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = V) |
14 | 13 | rexlimiva 3148 | . . . . 5 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 = V) |
15 | vex 3479 | . . . . 5 ⊢ 𝑦 ∈ V | |
16 | 14, 15 | eqeltrrdi 2843 | . . . 4 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → V ∈ V) |
17 | 4, 16 | mto 196 | . . 3 ⊢ ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 |
18 | 17 | abf 4403 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥} = ∅ |
19 | 3, 18 | eqtri 2761 | 1 ⊢ (fi‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3071 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 ∩ cint 4951 ‘cfv 6544 Fincfn 8939 ficfi 9405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-fi 9406 |
This theorem is referenced by: fieq0 9416 firest 17378 restbas 22662 |
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