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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 5213 | . . 3 ⊢ ∅ ∈ V | |
2 | fival 8878 | . . 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 5221 | . . . 4 ⊢ ¬ V ∈ V | |
5 | id 22 | . . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 → 𝑦 = ∩ 𝑥) | |
6 | elinel1 4174 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅) | |
7 | elpwi 4550 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅) | |
8 | ss0 4354 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
9 | 6, 7, 8 | 3syl 18 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅) |
10 | 9 | inteqd 4883 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = ∩ ∅) |
11 | int0 4892 | . . . . . . . 8 ⊢ ∩ ∅ = V | |
12 | 10, 11 | syl6eq 2874 | . . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = V) |
13 | 5, 12 | sylan9eqr 2880 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = V) |
14 | 13 | rexlimiva 3283 | . . . . 5 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 = V) |
15 | vex 3499 | . . . . 5 ⊢ 𝑦 ∈ V | |
16 | 14, 15 | eqeltrrdi 2924 | . . . 4 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → V ∈ V) |
17 | 4, 16 | mto 199 | . . 3 ⊢ ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 |
18 | 17 | abf 4358 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥} = ∅ |
19 | 3, 18 | eqtri 2846 | 1 ⊢ (fi‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {cab 2801 ∃wrex 3141 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 ∩ cint 4878 ‘cfv 6357 Fincfn 8511 ficfi 8876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-fi 8877 |
This theorem is referenced by: fieq0 8887 firest 16708 restbas 21768 |
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