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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 9679 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴))) | |
2 | reldom 8127 | . . . . . . 7 ⊢ Rel ≼ | |
3 | 2 | brrelex2i 5316 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
4 | xpexg 7125 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V) | |
5 | 3, 3, 4 | syl2anc 696 | . . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ∈ V) |
6 | cdadom3 9202 | . . . . 5 ⊢ (((𝐴 × 𝐴) ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ≼ ((𝐴 × 𝐴) +𝑐 𝐴)) | |
7 | 5, 3, 6 | syl2anc 696 | . . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≼ ((𝐴 × 𝐴) +𝑐 𝐴)) |
8 | cdacomen 9195 | . . . 4 ⊢ ((𝐴 × 𝐴) +𝑐 𝐴) ≈ (𝐴 +𝑐 (𝐴 × 𝐴)) | |
9 | domentr 8180 | . . . 4 ⊢ (((𝐴 × 𝐴) ≼ ((𝐴 × 𝐴) +𝑐 𝐴) ∧ ((𝐴 × 𝐴) +𝑐 𝐴) ≈ (𝐴 +𝑐 (𝐴 × 𝐴))) → (𝐴 × 𝐴) ≼ (𝐴 +𝑐 (𝐴 × 𝐴))) | |
10 | 7, 8, 9 | sylancl 697 | . . 3 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≼ (𝐴 +𝑐 (𝐴 × 𝐴))) |
11 | domtr 8174 | . . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≼ (𝐴 +𝑐 (𝐴 × 𝐴))) → 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴))) | |
12 | 11 | expcom 450 | . . 3 ⊢ ((𝐴 × 𝐴) ≼ (𝐴 +𝑐 (𝐴 × 𝐴)) → (𝒫 𝐴 ≼ (𝐴 × 𝐴) → 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))) |
13 | 10, 12 | syl 17 | . 2 ⊢ (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 × 𝐴) → 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))) |
14 | 1, 13 | mtod 189 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2139 Vcvv 3340 𝒫 cpw 4302 class class class wbr 4804 × cxp 5264 (class class class)co 6813 ωcom 7230 ≈ cen 8118 ≼ cdom 8119 +𝑐 ccda 9181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-seqom 7712 df-1o 7729 df-2o 7730 df-oadd 7733 df-omul 7734 df-oexp 7735 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-oi 8580 df-har 8628 df-cnf 8732 df-card 8955 df-cda 9182 |
This theorem is referenced by: gchxpidm 9683 |
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