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Mirrors > Home > MPE Home > Th. List > pwdjundom | Structured version Visualization version GIF version |
Description: The powerset of a Dedekind-infinite set does not inject into its cardinal sum with itself. (Contributed by Mario Carneiro, 31-May-2015.) |
Ref | Expression |
---|---|
pwdjundom | ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwxpndom2 10609 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
2 | df1o2 8423 | . . . . . . . 8 ⊢ 1o = {∅} | |
3 | 2 | xpeq1i 5663 | . . . . . . 7 ⊢ (1o × 𝐴) = ({∅} × 𝐴) |
4 | 0ex 5268 | . . . . . . . 8 ⊢ ∅ ∈ V | |
5 | reldom 8895 | . . . . . . . . 9 ⊢ Rel ≼ | |
6 | 5 | brrelex2i 5693 | . . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
7 | xpsnen2g 9015 | . . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
8 | 4, 6, 7 | sylancr 588 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
9 | 3, 8 | eqbrtrid 5144 | . . . . . 6 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≈ 𝐴) |
10 | 9 | ensymd 8951 | . . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (1o × 𝐴)) |
11 | omex 9587 | . . . . . . . 8 ⊢ ω ∈ V | |
12 | ordom 7816 | . . . . . . . . 9 ⊢ Ord ω | |
13 | 1onn 8590 | . . . . . . . . 9 ⊢ 1o ∈ ω | |
14 | ordelss 6337 | . . . . . . . . 9 ⊢ ((Ord ω ∧ 1o ∈ ω) → 1o ⊆ ω) | |
15 | 12, 13, 14 | mp2an 691 | . . . . . . . 8 ⊢ 1o ⊆ ω |
16 | ssdomg 8946 | . . . . . . . 8 ⊢ (ω ∈ V → (1o ⊆ ω → 1o ≼ ω)) | |
17 | 11, 15, 16 | mp2 9 | . . . . . . 7 ⊢ 1o ≼ ω |
18 | domtr 8953 | . . . . . . 7 ⊢ ((1o ≼ ω ∧ ω ≼ 𝐴) → 1o ≼ 𝐴) | |
19 | 17, 18 | mpan 689 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 1o ≼ 𝐴) |
20 | xpdom1g 9019 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 1o ≼ 𝐴) → (1o × 𝐴) ≼ (𝐴 × 𝐴)) | |
21 | 6, 19, 20 | syl2anc 585 | . . . . 5 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≼ (𝐴 × 𝐴)) |
22 | endomtr 8958 | . . . . 5 ⊢ ((𝐴 ≈ (1o × 𝐴) ∧ (1o × 𝐴) ≼ (𝐴 × 𝐴)) → 𝐴 ≼ (𝐴 × 𝐴)) | |
23 | 10, 21, 22 | syl2anc 585 | . . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 × 𝐴)) |
24 | djudom2 10127 | . . . 4 ⊢ ((𝐴 ≼ (𝐴 × 𝐴) ∧ 𝐴 ∈ V) → (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
25 | 23, 6, 24 | syl2anc 585 | . . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) |
26 | domtr 8953 | . . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴) ∧ (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
27 | 26 | expcom 415 | . . 3 ⊢ ((𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴)) → (𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))) |
28 | 25, 27 | syl 17 | . 2 ⊢ (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))) |
29 | 1, 28 | mtod 197 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 Vcvv 3447 ⊆ wss 3914 ∅c0 4286 𝒫 cpw 4564 {csn 4590 class class class wbr 5109 × cxp 5635 Ord word 6320 ωcom 7806 1oc1o 8409 ≈ cen 8886 ≼ cdom 8887 ⊔ cdju 9842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-seqom 8398 df-1o 8416 df-2o 8417 df-oadd 8420 df-omul 8421 df-oexp 8422 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-oi 9454 df-har 9501 df-cnf 9606 df-dju 9845 df-card 9883 |
This theorem is referenced by: gchdjuidm 10612 |
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