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| Mirrors > Home > MPE Home > Th. List > pwxpndom | Structured version Visualization version GIF version | ||
| Description: The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) |
| Ref | Expression |
|---|---|
| pwxpndom | ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 × 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pwxpndom2 10705 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
| 2 | reldom 8991 | . . . . . . 7 ⊢ Rel ≼ | |
| 3 | 2 | brrelex2i 5742 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
| 4 | 3, 3 | xpexd 7771 | . . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ∈ V) |
| 5 | djudoml 10225 | . . . . 5 ⊢ (((𝐴 × 𝐴) ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ≼ ((𝐴 × 𝐴) ⊔ 𝐴)) | |
| 6 | 4, 3, 5 | syl2anc 584 | . . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≼ ((𝐴 × 𝐴) ⊔ 𝐴)) |
| 7 | djucomen 10218 | . . . . 5 ⊢ (((𝐴 × 𝐴) ∈ V ∧ 𝐴 ∈ V) → ((𝐴 × 𝐴) ⊔ 𝐴) ≈ (𝐴 ⊔ (𝐴 × 𝐴))) | |
| 8 | 4, 3, 7 | syl2anc 584 | . . . 4 ⊢ (ω ≼ 𝐴 → ((𝐴 × 𝐴) ⊔ 𝐴) ≈ (𝐴 ⊔ (𝐴 × 𝐴))) |
| 9 | domentr 9053 | . . . 4 ⊢ (((𝐴 × 𝐴) ≼ ((𝐴 × 𝐴) ⊔ 𝐴) ∧ ((𝐴 × 𝐴) ⊔ 𝐴) ≈ (𝐴 ⊔ (𝐴 × 𝐴))) → (𝐴 × 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
| 10 | 6, 8, 9 | syl2anc 584 | . . 3 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) |
| 11 | domtr 9047 | . . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴))) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | |
| 12 | 11 | expcom 413 | . . 3 ⊢ ((𝐴 × 𝐴) ≼ (𝐴 ⊔ (𝐴 × 𝐴)) → (𝒫 𝐴 ≼ (𝐴 × 𝐴) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))) |
| 13 | 10, 12 | syl 17 | . 2 ⊢ (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 × 𝐴) → 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))) |
| 14 | 1, 13 | 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 𝒫 cpw 4600 class class class wbr 5143 × cxp 5683 ωcom 7887 ≈ 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: gchxpidm 10709 |
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