Theorem pwxpndom2 9487

Description: The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Assertion

Ref	Expression
pwxpndom2	⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))

Proof of Theorem pwxpndom2

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwfseq 9486	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
2		reldom 7961	. . . . . . 7 ⊢ Rel ≼
3	2	brrelex2i 5159	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		oveq1 6657	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ↑𝑚 1𝑜) = (𝐴 ↑𝑚 1𝑜))
5		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
6	4, 5	breq12d 4666	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ↑𝑚 1𝑜) ≈ 𝑥 ↔ (𝐴 ↑𝑚 1𝑜) ≈ 𝐴))
7		df1o2 7572	. . . . . . . . 9 ⊢ 1𝑜 = {∅}
8	7	oveq2i 6661	. . . . . . . 8 ⊢ (𝑥 ↑𝑚 1𝑜) = (𝑥 ↑𝑚 {∅})
9		vex 3203	. . . . . . . . 9 ⊢ 𝑥 ∈ V
10		0ex 4790	. . . . . . . . 9 ⊢ ∅ ∈ V
11	9, 10	mapsnen 8035	. . . . . . . 8 ⊢ (𝑥 ↑𝑚 {∅}) ≈ 𝑥
12	8, 11	eqbrtri 4674	. . . . . . 7 ⊢ (𝑥 ↑𝑚 1𝑜) ≈ 𝑥
13	6, 12	vtoclg 3266	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 1𝑜) ≈ 𝐴)
14		ensym 8005	. . . . . 6 ⊢ ((𝐴 ↑𝑚 1𝑜) ≈ 𝐴 → 𝐴 ≈ (𝐴 ↑𝑚 1𝑜))
15	3, 13, 14	3syl 18	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 ↑𝑚 1𝑜))
16		map2xp 8130	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 2𝑜) ≈ (𝐴 × 𝐴))
17		ensym 8005	. . . . . 6 ⊢ ((𝐴 ↑𝑚 2𝑜) ≈ (𝐴 × 𝐴) → (𝐴 × 𝐴) ≈ (𝐴 ↑𝑚 2𝑜))
18	3, 16, 17	3syl 18	. . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ (𝐴 ↑𝑚 2𝑜))
19		elmapi 7879	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 1𝑜) → 𝑥:1𝑜⟶𝐴)
20		fdm 6051	. . . . . . . . . . 11 ⊢ (𝑥:1𝑜⟶𝐴 → dom 𝑥 = 1𝑜)
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 1𝑜) → dom 𝑥 = 1𝑜)
22	21	adantr 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)) → dom 𝑥 = 1𝑜)
23		1onn 7719	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ ω
24	23	elexi 3213	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ V
25	24	sucid 5804	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ suc 1𝑜
26		df-2o 7561	. . . . . . . . . . . 12 ⊢ 2𝑜 = suc 1𝑜
27	25, 26	eleqtrri 2700	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ 2𝑜
28		1on 7567	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ On
29	28	onirri 5834	. . . . . . . . . . 11 ⊢ ¬ 1𝑜 ∈ 1𝑜
30		nelneq2 2726	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ 2𝑜 ∧ ¬ 1𝑜 ∈ 1𝑜) → ¬ 2𝑜 = 1𝑜)
31	27, 29, 30	mp2an 708	. . . . . . . . . 10 ⊢ ¬ 2𝑜 = 1𝑜
32		elmapi 7879	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 2𝑜) → 𝑥:2𝑜⟶𝐴)
33		fdm 6051	. . . . . . . . . . . . 13 ⊢ (𝑥:2𝑜⟶𝐴 → dom 𝑥 = 2𝑜)
34	32, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 2𝑜) → dom 𝑥 = 2𝑜)
35	34	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)) → dom 𝑥 = 2𝑜)
36	35	eqeq1d 2624	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)) → (dom 𝑥 = 1𝑜 ↔ 2𝑜 = 1𝑜))
37	31, 36	mtbiri 317	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)) → ¬ dom 𝑥 = 1𝑜)
38	22, 37	pm2.65i 185	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜))
39		elin 3796	. . . . . . . 8 ⊢ (𝑥 ∈ ((𝐴 ↑𝑚 1𝑜) ∩ (𝐴 ↑𝑚 2𝑜)) ↔ (𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)))
40	38, 39	mtbir 313	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ((𝐴 ↑𝑚 1𝑜) ∩ (𝐴 ↑𝑚 2𝑜))
41	40	a1i 11	. . . . . 6 ⊢ (ω ≼ 𝐴 → ¬ 𝑥 ∈ ((𝐴 ↑𝑚 1𝑜) ∩ (𝐴 ↑𝑚 2𝑜)))
42	41	eq0rdv 3979	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 ↑𝑚 1𝑜) ∩ (𝐴 ↑𝑚 2𝑜)) = ∅)
43		cdaenun 8996	. . . . 5 ⊢ ((𝐴 ≈ (𝐴 ↑𝑚 1𝑜) ∧ (𝐴 × 𝐴) ≈ (𝐴 ↑𝑚 2𝑜) ∧ ((𝐴 ↑𝑚 1𝑜) ∩ (𝐴 ↑𝑚 2𝑜)) = ∅) → (𝐴 +𝑐 (𝐴 × 𝐴)) ≈ ((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)))
44	15, 18, 42, 43	syl3anc 1326	. . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 (𝐴 × 𝐴)) ≈ ((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)))
45		omex 8540	. . . . . 6 ⊢ ω ∈ V
46		ovex 6678	. . . . . 6 ⊢ (𝐴 ↑𝑚 𝑛) ∈ V
47	45, 46	iunex 7147	. . . . 5 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∈ V
48		oveq2 6658	. . . . . . . 8 ⊢ (𝑛 = 1𝑜 → (𝐴 ↑𝑚 𝑛) = (𝐴 ↑𝑚 1𝑜))
49	48	ssiun2s 4564	. . . . . . 7 ⊢ (1𝑜 ∈ ω → (𝐴 ↑𝑚 1𝑜) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
50	23, 49	ax-mp 5	. . . . . 6 ⊢ (𝐴 ↑𝑚 1𝑜) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)
51		2onn 7720	. . . . . . 7 ⊢ 2𝑜 ∈ ω
52		oveq2 6658	. . . . . . . 8 ⊢ (𝑛 = 2𝑜 → (𝐴 ↑𝑚 𝑛) = (𝐴 ↑𝑚 2𝑜))
53	52	ssiun2s 4564	. . . . . . 7 ⊢ (2𝑜 ∈ ω → (𝐴 ↑𝑚 2𝑜) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
54	51, 53	ax-mp 5	. . . . . 6 ⊢ (𝐴 ↑𝑚 2𝑜) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)
55	50, 54	unssi 3788	. . . . 5 ⊢ ((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)
56		ssdomg 8001	. . . . 5 ⊢ (∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∈ V → (((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) → ((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)))
57	47, 55, 56	mp2 9	. . . 4 ⊢ ((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)
58		endomtr 8014	. . . 4 ⊢ (((𝐴 +𝑐 (𝐴 × 𝐴)) ≈ ((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)) ∧ ((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)) → (𝐴 +𝑐 (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
59	44, 57, 58	sylancl 694	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
60		domtr 8009	. . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)) ∧ (𝐴 +𝑐 (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
61	60	expcom 451	. . 3 ⊢ ((𝐴 +𝑐 (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) → (𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)))
62	59, 61	syl 17	. 2 ⊢ (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)))
63	1, 62	mtod 189	1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ ciun 4520   class class class wbr 4653   × cxp 5112  dom cdm 5114  suc csuc 5725  ⟶wf 5884  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   ↑_𝑚 cmap 7857   ≈ cen 7952   ≼ cdom 7953   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-har 8463  df-cnf 8559  df-card 8765  df-cda 8990

This theorem is referenced by:  pwxpndom  9488  pwcdandom  9489