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Mirrors > Home > MPE Home > Th. List > pwxpndom | Structured version Visualization version GIF version |
Description: The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) |
Ref | Expression |
---|---|
pwxpndom | ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 × 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwxpndom2 9885 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
2 | reldom 8312 | . . . . . . 7 ⊢ Rel ≼ | |
3 | 2 | brrelex2i 5459 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
4 | 3, 3 | xpexd 7291 | . . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ∈ V) |
5 | djudoml 9408 | . . . . 5 ⊢ (((𝐴 × 𝐴) ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ≼ ((𝐴 × 𝐴) ⊔ 𝐴)) | |
6 | 4, 3, 5 | syl2anc 576 | . . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≼ ((𝐴 × 𝐴) ⊔ 𝐴)) |
7 | djucomen 9401 | . . . . 5 ⊢ (((𝐴 × 𝐴) ∈ V ∧ 𝐴 ∈ V) → ((𝐴 × 𝐴) ⊔ 𝐴) ≈ (𝐴 ⊔ (𝐴 × 𝐴))) | |
8 | 4, 3, 7 | syl2anc 576 | . . . 4 ⊢ (ω ≼ 𝐴 → ((𝐴 × 𝐴) ⊔ 𝐴) ≈ (𝐴 ⊔ (𝐴 × 𝐴))) |
9 | domentr 8365 | . . . 4 ⊢ (((𝐴 × 𝐴) ≼ ((𝐴 × 𝐴) ⊔ 𝐴) ∧ ((𝐴 × 𝐴) ⊔ 𝐴) ≈ (𝐴 ⊔ (𝐴 × 𝐴))) → (𝐴 × 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
10 | 6, 8, 9 | syl2anc 576 | . . 3 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) |
11 | domtr 8359 | . . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
12 | 11 | expcom 406 | . . 3 ⊢ ((𝐴 × 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴)) → (𝒫 𝐴 ≼ (𝐴 × 𝐴) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))) |
13 | 10, 12 | syl 17 | . 2 ⊢ (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 × 𝐴) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))) |
14 | 1, 13 | mtod 190 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2050 Vcvv 3415 𝒫 cpw 4422 class class class wbr 4929 × cxp 5405 ωcom 7396 ≈ cen 8303 ≼ cdom 8304 ⊔ cdju 9121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-seqom 7887 df-1o 7905 df-2o 7906 df-oadd 7909 df-omul 7910 df-oexp 7911 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-oi 8769 df-har 8817 df-cnf 8919 df-dju 9124 df-card 9162 |
This theorem is referenced by: gchxpidm 9889 |
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