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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 10262 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
2 | df1o2 8203 | . . . . . . . 8 ⊢ 1o = {∅} | |
3 | 2 | xpeq1i 5566 | . . . . . . 7 ⊢ (1o × 𝐴) = ({∅} × 𝐴) |
4 | 0ex 5189 | . . . . . . . 8 ⊢ ∅ ∈ V | |
5 | reldom 8621 | . . . . . . . . 9 ⊢ Rel ≼ | |
6 | 5 | brrelex2i 5595 | . . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
7 | xpsnen2g 8727 | . . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
8 | 4, 6, 7 | sylancr 590 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
9 | 3, 8 | eqbrtrid 5078 | . . . . . 6 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≈ 𝐴) |
10 | 9 | ensymd 8668 | . . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (1o × 𝐴)) |
11 | omex 9247 | . . . . . . . 8 ⊢ ω ∈ V | |
12 | ordom 7643 | . . . . . . . . 9 ⊢ Ord ω | |
13 | 1onn 8356 | . . . . . . . . 9 ⊢ 1o ∈ ω | |
14 | ordelss 6218 | . . . . . . . . 9 ⊢ ((Ord ω ∧ 1o ∈ ω) → 1o ⊆ ω) | |
15 | 12, 13, 14 | mp2an 692 | . . . . . . . 8 ⊢ 1o ⊆ ω |
16 | ssdomg 8663 | . . . . . . . 8 ⊢ (ω ∈ V → (1o ⊆ ω → 1o ≼ ω)) | |
17 | 11, 15, 16 | mp2 9 | . . . . . . 7 ⊢ 1o ≼ ω |
18 | domtr 8670 | . . . . . . 7 ⊢ ((1o ≼ ω ∧ ω ≼ 𝐴) → 1o ≼ 𝐴) | |
19 | 17, 18 | mpan 690 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 1o ≼ 𝐴) |
20 | xpdom1g 8731 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 1o ≼ 𝐴) → (1o × 𝐴) ≼ (𝐴 × 𝐴)) | |
21 | 6, 19, 20 | syl2anc 587 | . . . . 5 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≼ (𝐴 × 𝐴)) |
22 | endomtr 8675 | . . . . 5 ⊢ ((𝐴 ≈ (1o × 𝐴) ∧ (1o × 𝐴) ≼ (𝐴 × 𝐴)) → 𝐴 ≼ (𝐴 × 𝐴)) | |
23 | 10, 21, 22 | syl2anc 587 | . . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 × 𝐴)) |
24 | djudom2 9780 | . . . 4 ⊢ ((𝐴 ≼ (𝐴 × 𝐴) ∧ 𝐴 ∈ V) → (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
25 | 23, 6, 24 | syl2anc 587 | . . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) |
26 | domtr 8670 | . . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴) ∧ (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
27 | 26 | expcom 417 | . . 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 201 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2110 Vcvv 3401 ⊆ wss 3857 ∅c0 4227 𝒫 cpw 4503 {csn 4531 class class class wbr 5043 × cxp 5538 Ord word 6201 ωcom 7633 1oc1o 8184 ≈ cen 8612 ≼ cdom 8613 ⊔ cdju 9497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-seqom 8173 df-1o 8191 df-2o 8192 df-oadd 8195 df-omul 8196 df-oexp 8197 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-oi 9115 df-har 9162 df-cnf 9266 df-dju 9500 df-card 9538 |
This theorem is referenced by: gchdjuidm 10265 |
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