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Mirrors > Home > MPE Home > Th. List > ixpprc | Structured version Visualization version 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 | neq0 4358 | . . 3 ⊢ (¬ X𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) | |
2 | ixpfn 8942 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴) | |
3 | fndm 6672 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴) | |
4 | vex 3482 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
5 | 4 | dmex 7932 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
6 | 3, 5 | eqeltrrdi 2848 | . . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V) |
7 | 2, 6 | syl 17 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V) |
8 | 7 | exlimiv 1928 | . . 3 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V) |
9 | 1, 8 | sylbi 217 | . 2 ⊢ (¬ X𝑥 ∈ 𝐴 𝐵 = ∅ → 𝐴 ∈ V) |
10 | 9 | con1i 147 | 1 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 dom cdm 5689 Fn wfn 6558 Xcixp 8936 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-ixp 8937 |
This theorem is referenced by: ixpexg 8961 ixpssmap2g 8966 ixpssmapg 8967 resixpfo 8975 |
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