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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 10087 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
2 | df1o2 8116 | . . . . . . . 8 ⊢ 1o = {∅} | |
3 | 2 | xpeq1i 5581 | . . . . . . 7 ⊢ (1o × 𝐴) = ({∅} × 𝐴) |
4 | 0ex 5211 | . . . . . . . 8 ⊢ ∅ ∈ V | |
5 | reldom 8515 | . . . . . . . . 9 ⊢ Rel ≼ | |
6 | 5 | brrelex2i 5609 | . . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
7 | xpsnen2g 8610 | . . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
8 | 4, 6, 7 | sylancr 589 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
9 | 3, 8 | eqbrtrid 5101 | . . . . . 6 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≈ 𝐴) |
10 | 9 | ensymd 8560 | . . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (1o × 𝐴)) |
11 | omex 9106 | . . . . . . . 8 ⊢ ω ∈ V | |
12 | ordom 7589 | . . . . . . . . 9 ⊢ Ord ω | |
13 | 1onn 8265 | . . . . . . . . 9 ⊢ 1o ∈ ω | |
14 | ordelss 6207 | . . . . . . . . 9 ⊢ ((Ord ω ∧ 1o ∈ ω) → 1o ⊆ ω) | |
15 | 12, 13, 14 | mp2an 690 | . . . . . . . 8 ⊢ 1o ⊆ ω |
16 | ssdomg 8555 | . . . . . . . 8 ⊢ (ω ∈ V → (1o ⊆ ω → 1o ≼ ω)) | |
17 | 11, 15, 16 | mp2 9 | . . . . . . 7 ⊢ 1o ≼ ω |
18 | domtr 8562 | . . . . . . 7 ⊢ ((1o ≼ ω ∧ ω ≼ 𝐴) → 1o ≼ 𝐴) | |
19 | 17, 18 | mpan 688 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 1o ≼ 𝐴) |
20 | xpdom1g 8614 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 1o ≼ 𝐴) → (1o × 𝐴) ≼ (𝐴 × 𝐴)) | |
21 | 6, 19, 20 | syl2anc 586 | . . . . 5 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≼ (𝐴 × 𝐴)) |
22 | endomtr 8567 | . . . . 5 ⊢ ((𝐴 ≈ (1o × 𝐴) ∧ (1o × 𝐴) ≼ (𝐴 × 𝐴)) → 𝐴 ≼ (𝐴 × 𝐴)) | |
23 | 10, 21, 22 | syl2anc 586 | . . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 × 𝐴)) |
24 | djudom2 9609 | . . . 4 ⊢ ((𝐴 ≼ (𝐴 × 𝐴) ∧ 𝐴 ∈ V) → (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
25 | 23, 6, 24 | syl2anc 586 | . . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) |
26 | domtr 8562 | . . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴) ∧ (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
27 | 26 | expcom 416 | . . 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 200 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 {csn 4567 class class class wbr 5066 × cxp 5553 Ord word 6190 ωcom 7580 1oc1o 8095 ≈ cen 8506 ≼ cdom 8507 ⊔ cdju 9327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-seqom 8084 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-oexp 8108 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-har 9022 df-cnf 9125 df-dju 9330 df-card 9368 |
This theorem is referenced by: gchdjuidm 10090 |
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