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Mirrors > Home > MPE Home > Th. List > pwcdandom | 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 |
---|---|
pwcdandom | ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwxpndom2 9366 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴))) | |
2 | df1o2 7459 | . . . . . . 7 ⊢ 1𝑜 = {∅} | |
3 | 2 | xpeq2i 5060 | . . . . . 6 ⊢ (𝐴 × 1𝑜) = (𝐴 × {∅}) |
4 | reldom 7847 | . . . . . . . 8 ⊢ Rel ≼ | |
5 | 4 | brrelex2i 5083 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
6 | 0ex 4718 | . . . . . . 7 ⊢ ∅ ∈ V | |
7 | xpsneng 7930 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴) | |
8 | 5, 6, 7 | sylancl 693 | . . . . . 6 ⊢ (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ 𝐴) |
9 | 3, 8 | syl5eqbr 4618 | . . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 × 1𝑜) ≈ 𝐴) |
10 | 9 | ensymd 7893 | . . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 × 1𝑜)) |
11 | omex 8423 | . . . . . . 7 ⊢ ω ∈ V | |
12 | ordom 6966 | . . . . . . . 8 ⊢ Ord ω | |
13 | 1onn 7606 | . . . . . . . 8 ⊢ 1𝑜 ∈ ω | |
14 | ordelss 5656 | . . . . . . . 8 ⊢ ((Ord ω ∧ 1𝑜 ∈ ω) → 1𝑜 ⊆ ω) | |
15 | 12, 13, 14 | mp2an 704 | . . . . . . 7 ⊢ 1𝑜 ⊆ ω |
16 | ssdomg 7887 | . . . . . . 7 ⊢ (ω ∈ V → (1𝑜 ⊆ ω → 1𝑜 ≼ ω)) | |
17 | 11, 15, 16 | mp2 9 | . . . . . 6 ⊢ 1𝑜 ≼ ω |
18 | domtr 7895 | . . . . . 6 ⊢ ((1𝑜 ≼ ω ∧ ω ≼ 𝐴) → 1𝑜 ≼ 𝐴) | |
19 | 17, 18 | mpan 702 | . . . . 5 ⊢ (ω ≼ 𝐴 → 1𝑜 ≼ 𝐴) |
20 | xpdom2g 7941 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ≼ 𝐴) → (𝐴 × 1𝑜) ≼ (𝐴 × 𝐴)) | |
21 | 5, 19, 20 | syl2anc 691 | . . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 × 1𝑜) ≼ (𝐴 × 𝐴)) |
22 | endomtr 7900 | . . . 4 ⊢ ((𝐴 ≈ (𝐴 × 1𝑜) ∧ (𝐴 × 1𝑜) ≼ (𝐴 × 𝐴)) → 𝐴 ≼ (𝐴 × 𝐴)) | |
23 | 10, 21, 22 | syl2anc 691 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 × 𝐴)) |
24 | cdadom2 8892 | . . 3 ⊢ (𝐴 ≼ (𝐴 × 𝐴) → (𝐴 +𝑐 𝐴) ≼ (𝐴 +𝑐 (𝐴 × 𝐴))) | |
25 | domtr 7895 | . . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 +𝑐 𝐴) ∧ (𝐴 +𝑐 𝐴) ≼ (𝐴 +𝑐 (𝐴 × 𝐴))) → 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴))) | |
26 | 25 | expcom 450 | . . 3 ⊢ ((𝐴 +𝑐 𝐴) ≼ (𝐴 +𝑐 (𝐴 × 𝐴)) → (𝒫 𝐴 ≼ (𝐴 +𝑐 𝐴) → 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))) |
27 | 23, 24, 26 | 3syl 18 | . 2 ⊢ (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 +𝑐 𝐴) → 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))) |
28 | 1, 27 | mtod 188 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 × cxp 5036 Ord word 5639 (class class class)co 6549 ωcom 6957 1𝑜c1o 7440 ≈ cen 7838 ≼ cdom 7839 +𝑐 ccda 8872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-oexp 7453 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-har 8346 df-cnf 8442 df-card 8648 df-cda 8873 |
This theorem is referenced by: gchcdaidm 9369 |
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