Theorem pwxpndom2 9366

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 9365	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
2		reldom 7847	. . . . . . 7 ⊢ Rel ≼
3	2	brrelex2i 5083	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		oveq1 6556	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ↑𝑚 1𝑜) = (𝐴 ↑𝑚 1𝑜))
5		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
6	4, 5	breq12d 4596	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ↑𝑚 1𝑜) ≈ 𝑥 ↔ (𝐴 ↑𝑚 1𝑜) ≈ 𝐴))
7		df1o2 7459	. . . . . . . . 9 ⊢ 1𝑜 = {∅}
8	7	oveq2i 6560	. . . . . . . 8 ⊢ (𝑥 ↑𝑚 1𝑜) = (𝑥 ↑𝑚 {∅})
9		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
10		0ex 4718	. . . . . . . . 9 ⊢ ∅ ∈ V
11	9, 10	mapsnen 7920	. . . . . . . 8 ⊢ (𝑥 ↑𝑚 {∅}) ≈ 𝑥
12	8, 11	eqbrtri 4604	. . . . . . 7 ⊢ (𝑥 ↑𝑚 1𝑜) ≈ 𝑥
13	6, 12	vtoclg 3239	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 1𝑜) ≈ 𝐴)
14		ensym 7891	. . . . . 6 ⊢ ((𝐴 ↑𝑚 1𝑜) ≈ 𝐴 → 𝐴 ≈ (𝐴 ↑𝑚 1𝑜))
15	3, 13, 14	3syl 18	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 ↑𝑚 1𝑜))
16		map2xp 8015	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 2𝑜) ≈ (𝐴 × 𝐴))
17		ensym 7891	. . . . . 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 7765	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 1𝑜) → 𝑥:1𝑜⟶𝐴)
20		fdm 5964	. . . . . . . . . . 11 ⊢ (𝑥:1𝑜⟶𝐴 → dom 𝑥 = 1𝑜)
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 1𝑜) → dom 𝑥 = 1𝑜)
22	21	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)) → dom 𝑥 = 1𝑜)
23		1onn 7606	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ ω
24	23	elexi 3186	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ V
25	24	sucid 5721	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ suc 1𝑜
26		df-2o 7448	. . . . . . . . . . . 12 ⊢ 2𝑜 = suc 1𝑜
27	25, 26	eleqtrri 2687	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ 2𝑜
28		1on 7454	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ On
29	28	onirri 5751	. . . . . . . . . . 11 ⊢ ¬ 1𝑜 ∈ 1𝑜
30		nelneq2 2713	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ 2𝑜 ∧ ¬ 1𝑜 ∈ 1𝑜) → ¬ 2𝑜 = 1𝑜)
31	27, 29, 30	mp2an 704	. . . . . . . . . 10 ⊢ ¬ 2𝑜 = 1𝑜
32		elmapi 7765	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 2𝑜) → 𝑥:2𝑜⟶𝐴)
33		fdm 5964	. . . . . . . . . . . . 13 ⊢ (𝑥:2𝑜⟶𝐴 → dom 𝑥 = 2𝑜)
34	32, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 2𝑜) → dom 𝑥 = 2𝑜)
35	34	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)) → dom 𝑥 = 2𝑜)
36	35	eqeq1d 2612	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)) → (dom 𝑥 = 1𝑜 ↔ 2𝑜 = 1𝑜))
37	31, 36	mtbiri 316	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)) → ¬ dom 𝑥 = 1𝑜)
38	22, 37	pm2.65i 184	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜))
39		elin 3758	. . . . . . . 8 ⊢ (𝑥 ∈ ((𝐴 ↑𝑚 1𝑜) ∩ (𝐴 ↑𝑚 2𝑜)) ↔ (𝑥 ∈ (𝐴 ↑𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2𝑜)))
40	38, 39	mtbir 312	. . . . . . 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 3931	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 ↑𝑚 1𝑜) ∩ (𝐴 ↑𝑚 2𝑜)) = ∅)
43		cdaenun 8879	. . . . 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 1318	. . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 (𝐴 × 𝐴)) ≈ ((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)))
45		omex 8423	. . . . . 6 ⊢ ω ∈ V
46		ovex 6577	. . . . . 6 ⊢ (𝐴 ↑𝑚 𝑛) ∈ V
47	45, 46	iunex 7039	. . . . 5 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∈ V
48		oveq2 6557	. . . . . . . 8 ⊢ (𝑛 = 1𝑜 → (𝐴 ↑𝑚 𝑛) = (𝐴 ↑𝑚 1𝑜))
49	48	ssiun2s 4500	. . . . . . 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 7607	. . . . . . 7 ⊢ 2𝑜 ∈ ω
52		oveq2 6557	. . . . . . . 8 ⊢ (𝑛 = 2𝑜 → (𝐴 ↑𝑚 𝑛) = (𝐴 ↑𝑚 2𝑜))
53	52	ssiun2s 4500	. . . . . . 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 3750	. . . . 5 ⊢ ((𝐴 ↑𝑚 1𝑜) ∪ (𝐴 ↑𝑚 2𝑜)) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)
56		ssdomg 7887	. . . . 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 7900	. . . 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 693	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
60		domtr 7895	. . . 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 450	. . 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 188	1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ ciun 4455   class class class wbr 4583   × cxp 5036  dom cdm 5038  suc csuc 5642  ⟶wf 5800  (class class class)co 6549  ωcom 6957  1_𝑜c1o 7440  2_𝑜c2o 7441   ↑_𝑚 cmap 7744   ≈ 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:  pwxpndom  9367  pwcdandom  9368