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Mirrors > Home > ILE Home > Th. List > ixpprc | GIF version |
Description: A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain 𝐴, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.) |
Ref | Expression |
---|---|
ixpprc | ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpfn 6591 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴) | |
2 | fndm 5217 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴) | |
3 | vex 2684 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
4 | 3 | dmex 4800 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
5 | 2, 4 | eqeltrrdi 2229 | . . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V) |
6 | 1, 5 | syl 14 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V) |
7 | 6 | exlimiv 1577 | . . 3 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V) |
8 | 7 | con3i 621 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) |
9 | notm0 3378 | . 2 ⊢ (¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ X𝑥 ∈ 𝐴 𝐵 = ∅) | |
10 | 8, 9 | sylib 121 | 1 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2681 ∅c0 3358 dom cdm 4534 Fn wfn 5113 Xcixp 6585 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-ixp 6586 |
This theorem is referenced by: (None) |
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