|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 4351 | . . 3 ⊢ (¬ X𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) | |
| 2 | ixpfn 8944 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴) | |
| 3 | fndm 6670 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴) | |
| 4 | vex 3483 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
| 5 | 4 | dmex 7932 | . . . . . 6 ⊢ dom 𝑓 ∈ V | 
| 6 | 3, 5 | eqeltrrdi 2849 | . . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V) | 
| 7 | 2, 6 | syl 17 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V) | 
| 8 | 7 | exlimiv 1929 | . . 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 1539 ∃wex 1778 ∈ wcel 2107 Vcvv 3479 ∅c0 4332 dom cdm 5684 Fn wfn 6555 Xcixp 8938 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 df-ixp 8939 | 
| This theorem is referenced by: ixpexg 8963 ixpssmap2g 8968 ixpssmapg 8969 resixpfo 8977 | 
| Copyright terms: Public domain | W3C validator |