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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 9933 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
2 | df1o2 7967 | . . . . . . . 8 ⊢ 1o = {∅} | |
3 | 2 | xpeq1i 5469 | . . . . . . 7 ⊢ (1o × 𝐴) = ({∅} × 𝐴) |
4 | 0ex 5102 | . . . . . . . 8 ⊢ ∅ ∈ V | |
5 | reldom 8363 | . . . . . . . . 9 ⊢ Rel ≼ | |
6 | 5 | brrelex2i 5495 | . . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
7 | xpsnen2g 8457 | . . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
8 | 4, 6, 7 | sylancr 587 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
9 | 3, 8 | eqbrtrid 4997 | . . . . . 6 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≈ 𝐴) |
10 | 9 | ensymd 8408 | . . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (1o × 𝐴)) |
11 | omex 8952 | . . . . . . . 8 ⊢ ω ∈ V | |
12 | ordom 7445 | . . . . . . . . 9 ⊢ Ord ω | |
13 | 1onn 8115 | . . . . . . . . 9 ⊢ 1o ∈ ω | |
14 | ordelss 6082 | . . . . . . . . 9 ⊢ ((Ord ω ∧ 1o ∈ ω) → 1o ⊆ ω) | |
15 | 12, 13, 14 | mp2an 688 | . . . . . . . 8 ⊢ 1o ⊆ ω |
16 | ssdomg 8403 | . . . . . . . 8 ⊢ (ω ∈ V → (1o ⊆ ω → 1o ≼ ω)) | |
17 | 11, 15, 16 | mp2 9 | . . . . . . 7 ⊢ 1o ≼ ω |
18 | domtr 8410 | . . . . . . 7 ⊢ ((1o ≼ ω ∧ ω ≼ 𝐴) → 1o ≼ 𝐴) | |
19 | 17, 18 | mpan 686 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 1o ≼ 𝐴) |
20 | xpdom1g 8461 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 1o ≼ 𝐴) → (1o × 𝐴) ≼ (𝐴 × 𝐴)) | |
21 | 6, 19, 20 | syl2anc 584 | . . . . 5 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≼ (𝐴 × 𝐴)) |
22 | endomtr 8415 | . . . . 5 ⊢ ((𝐴 ≈ (1o × 𝐴) ∧ (1o × 𝐴) ≼ (𝐴 × 𝐴)) → 𝐴 ≼ (𝐴 × 𝐴)) | |
23 | 10, 21, 22 | syl2anc 584 | . . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 × 𝐴)) |
24 | djudom2 9455 | . . . 4 ⊢ ((𝐴 ≼ (𝐴 × 𝐴) ∧ 𝐴 ∈ V) → (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
25 | 23, 6, 24 | syl2anc 584 | . . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) |
26 | domtr 8410 | . . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴) ∧ (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
27 | 26 | expcom 414 | . . 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 199 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2081 Vcvv 3437 ⊆ wss 3859 ∅c0 4211 𝒫 cpw 4453 {csn 4472 class class class wbr 4962 × cxp 5441 Ord word 6065 ωcom 7436 1oc1o 7946 ≈ cen 8354 ≼ cdom 8355 ⊔ cdju 9173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-seqom 7935 df-1o 7953 df-2o 7954 df-oadd 7957 df-omul 7958 df-oexp 7959 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-oi 8820 df-har 8868 df-cnf 8971 df-dju 9176 df-card 9214 |
This theorem is referenced by: gchdjuidm 9936 |
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