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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 10705 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
| 2 | df1o2 8513 | . . . . . . . 8 ⊢ 1o = {∅} | |
| 3 | 2 | xpeq1i 5711 | . . . . . . 7 ⊢ (1o × 𝐴) = ({∅} × 𝐴) | 
| 4 | 0ex 5307 | . . . . . . . 8 ⊢ ∅ ∈ V | |
| 5 | reldom 8991 | . . . . . . . . 9 ⊢ Rel ≼ | |
| 6 | 5 | brrelex2i 5742 | . . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) | 
| 7 | xpsnen2g 9105 | . . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
| 8 | 4, 6, 7 | sylancr 587 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) | 
| 9 | 3, 8 | eqbrtrid 5178 | . . . . . 6 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≈ 𝐴) | 
| 10 | 9 | ensymd 9045 | . . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (1o × 𝐴)) | 
| 11 | omex 9683 | . . . . . . . 8 ⊢ ω ∈ V | |
| 12 | ordom 7897 | . . . . . . . . 9 ⊢ Ord ω | |
| 13 | 1onn 8678 | . . . . . . . . 9 ⊢ 1o ∈ ω | |
| 14 | ordelss 6400 | . . . . . . . . 9 ⊢ ((Ord ω ∧ 1o ∈ ω) → 1o ⊆ ω) | |
| 15 | 12, 13, 14 | mp2an 692 | . . . . . . . 8 ⊢ 1o ⊆ ω | 
| 16 | ssdomg 9040 | . . . . . . . 8 ⊢ (ω ∈ V → (1o ⊆ ω → 1o ≼ ω)) | |
| 17 | 11, 15, 16 | mp2 9 | . . . . . . 7 ⊢ 1o ≼ ω | 
| 18 | domtr 9047 | . . . . . . 7 ⊢ ((1o ≼ ω ∧ ω ≼ 𝐴) → 1o ≼ 𝐴) | |
| 19 | 17, 18 | mpan 690 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 1o ≼ 𝐴) | 
| 20 | xpdom1g 9109 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 1o ≼ 𝐴) → (1o × 𝐴) ≼ (𝐴 × 𝐴)) | |
| 21 | 6, 19, 20 | syl2anc 584 | . . . . 5 ⊢ (ω ≼ 𝐴 → (1o × 𝐴) ≼ (𝐴 × 𝐴)) | 
| 22 | endomtr 9052 | . . . . 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 10224 | . . . 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 9047 | . . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴) ∧ (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
| 27 | 26 | expcom 413 | . . 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 198 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 class class class wbr 5143 × cxp 5683 Ord word 6383 ωcom 7887 1oc1o 8499 ≈ cen 8982 ≼ cdom 8983 ⊔ cdju 9938 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seqom 8488 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-oexp 8512 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-har 9597 df-cnf 9702 df-dju 9941 df-card 9979 | 
| This theorem is referenced by: gchdjuidm 10708 | 
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