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Mirrors > Home > ILE Home > Th. List > fi0 | 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 4063 | . . 3 ⊢ ∅ ∈ V | |
2 | fival 6866 | . . 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 4068 | . . . 4 ⊢ ¬ V ∈ V | |
5 | id 19 | . . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 → 𝑦 = ∩ 𝑥) | |
6 | elinel1 3267 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅) | |
7 | elpwi 3524 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅) | |
8 | ss0 3408 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
9 | 6, 7, 8 | 3syl 17 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅) |
10 | 9 | inteqd 3784 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = ∩ ∅) |
11 | int0 3793 | . . . . . . . 8 ⊢ ∩ ∅ = V | |
12 | 10, 11 | eqtrdi 2189 | . . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = V) |
13 | 5, 12 | sylan9eqr 2195 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = V) |
14 | 13 | rexlimiva 2547 | . . . . 5 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 = V) |
15 | vex 2692 | . . . . 5 ⊢ 𝑦 ∈ V | |
16 | 14, 15 | eqeltrrdi 2232 | . . . 4 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → V ∈ V) |
17 | 4, 16 | mto 652 | . . 3 ⊢ ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 |
18 | 17 | abf 3411 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥} = ∅ |
19 | 3, 18 | eqtri 2161 | 1 ⊢ (fi‘∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 {cab 2126 ∃wrex 2418 Vcvv 2689 ∩ cin 3075 ⊆ wss 3076 ∅c0 3368 𝒫 cpw 3515 ∩ cint 3779 ‘cfv 5131 Fincfn 6642 ficfi 6864 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-er 6437 df-en 6643 df-fin 6645 df-fi 6865 |
This theorem is referenced by: fieq0 6872 |
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