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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 4347 | . . 3 ⊢ (¬ X𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) | |
2 | ixpfn 8926 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴) | |
3 | fndm 6660 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴) | |
4 | vex 3475 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
5 | 4 | dmex 7921 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
6 | 3, 5 | eqeltrrdi 2837 | . . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V) |
7 | 2, 6 | syl 17 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V) |
8 | 7 | exlimiv 1925 | . . 3 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V) |
9 | 1, 8 | sylbi 216 | . 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 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3471 ∅c0 4324 dom cdm 5680 Fn wfn 6546 Xcixp 8920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-iota 6503 df-fun 6553 df-fn 6554 df-fv 6559 df-ixp 8921 |
This theorem is referenced by: ixpexg 8945 ixpssmap2g 8950 ixpssmapg 8951 resixpfo 8959 |
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