Theorem pwxpndom2 10638

Description: The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) (Proof shortened by AV, 18-Jul-2022.)

Assertion

Ref	Expression
pwxpndom2	⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (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 10637	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
2		reldom 8937	. . . . . . 7 ⊢ Rel ≼
3	2	brrelex2i 5709	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		df1o2 8448	. . . . . . . 8 ⊢ 1o = {∅}
5	4	oveq2i 7411	. . . . . . 7 ⊢ (𝐴 ↑m 1o) = (𝐴 ↑m {∅})
6		id 23	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
7		0ex 5262	. . . . . . . . 9 ⊢ ∅ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∅ ∈ V)
9	6, 8	mapsnend 9021	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ↑m {∅}) ≈ 𝐴)
10	5, 9	eqbrtrid 5140	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 1o) ≈ 𝐴)
11		ensym 8988	. . . . . 6 ⊢ ((𝐴 ↑m 1o) ≈ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
12	3, 10, 11	3syl 19	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
13		map2xp 9123	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))
14		ensym 8988	. . . . . 6 ⊢ ((𝐴 ↑m 2o) ≈ (𝐴 × 𝐴) → (𝐴 × 𝐴) ≈ (𝐴 ↑m 2o))
15	3, 13, 14	3syl 19	. . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ (𝐴 ↑m 2o))
16		elmapi 8834	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ↑m 1o) → 𝑥:1o⟶𝐴)
17	16	fdmd 6706	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ↑m 1o) → dom 𝑥 = 1o)
18	17	adantr 485	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → dom 𝑥 = 1o)
19		1oex 8451	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
20	19	sucid 6434	. . . . . . . . . . . 12 ⊢ 1o ∈ suc 1o
21		df-2o 8442	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
22	20, 21	eleqtrri 2864	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
23		1on 8454	. . . . . . . . . . . 12 ⊢ 1o ∈ On
24	23	onirri 6464	. . . . . . . . . . 11 ⊢ ¬ 1o ∈ 1o
25		nelneq2 2890	. . . . . . . . . . 11 ⊢ ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
26	22, 24, 25	mp2an 704	. . . . . . . . . 10 ⊢ ¬ 2o = 1o
27		elmapi 8834	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ↑m 2o) → 𝑥:2o⟶𝐴)
28	27	fdmd 6706	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ↑m 2o) → dom 𝑥 = 2o)
29	28	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → dom 𝑥 = 2o)
30	29	eqeq1d 2767	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → (dom 𝑥 = 1o ↔ 2o = 1o))
31	26, 30	mtbiri 330	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → ¬ dom 𝑥 = 1o)
32	18, 31	pm2.65i 196	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o))
33		elin 3923	. . . . . . . 8 ⊢ (𝑥 ∈ ((𝐴 ↑m 1o) ∩ (𝐴 ↑m 2o)) ↔ (𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)))
34	32, 33	mtbir 326	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ((𝐴 ↑m 1o) ∩ (𝐴 ↑m 2o))
35	34	a1i 11	. . . . . 6 ⊢ (ω ≼ 𝐴 → ¬ 𝑥 ∈ ((𝐴 ↑m 1o) ∩ (𝐴 ↑m 2o)))
36	35	eq0rdv 4364	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 ↑m 1o) ∩ (𝐴 ↑m 2o)) = ∅)
37		djuenun 10142	. . . . 5 ⊢ ((𝐴 ≈ (𝐴 ↑m 1o) ∧ (𝐴 × 𝐴) ≈ (𝐴 ↑m 2o) ∧ ((𝐴 ↑m 1o) ∩ (𝐴 ↑m 2o)) = ∅) → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)))
38	12, 15, 36, 37	syl3anc 1394	. . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)))
39		omex 9600	. . . . . 6 ⊢ ω ∈ V
40		ovex 7433	. . . . . 6 ⊢ (𝐴 ↑m 𝑛) ∈ V
41	39, 40	iunex 7953	. . . . 5 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ V
42		1onn 8614	. . . . . . 7 ⊢ 1o ∈ ω
43		oveq2 7408	. . . . . . . 8 ⊢ (𝑛 = 1o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 1o))
44	43	ssiun2s 5009	. . . . . . 7 ⊢ (1o ∈ ω → (𝐴 ↑m 1o) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
45	42, 44	ax-mp 5	. . . . . 6 ⊢ (𝐴 ↑m 1o) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)
46		2onn 8616	. . . . . . 7 ⊢ 2o ∈ ω
47		oveq2 7408	. . . . . . . 8 ⊢ (𝑛 = 2o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 2o))
48	47	ssiun2s 5009	. . . . . . 7 ⊢ (2o ∈ ω → (𝐴 ↑m 2o) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
49	46, 48	ax-mp 5	. . . . . 6 ⊢ (𝐴 ↑m 2o) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)
50	45, 49	unssi 4146	. . . . 5 ⊢ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)
51		ssdomg 8985	. . . . 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)))
52	41, 50, 51	mp2 9	. . . 4 ⊢ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)
53		endomtr 8997	. . . 4 ⊢ (((𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)) ∧ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
54	38, 52, 53	sylancl 597	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
55		domtr 8992	. . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) ∧ (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
56	55	expcom 418	. . 3 ⊢ ((𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) → (𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
57	54, 56	syl 18	. 2 ⊢ (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
58	1, 57	mtod 201	1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  {csn 4585  ∪ ciun 4952   class class class wbr 5105   × cxp 5650  dom cdm 5652  suc csuc 6352  (class class class)co 7400  ωcom 7850  1_oc1o 8434  2_oc2o 8435   ↑_m cmap 8812   ≈ cen 8928   ≼ cdom 8929   ⊔ cdju 9872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-seqom 8423  df-1o 8441  df-2o 8442  df-oadd 8445  df-omul 8446  df-oexp 8447  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-oi 9460  df-har 9507  df-cnf 9619  df-dju 9875  df-card 9913

This theorem is referenced by:  pwxpndom  10639  pwdjundom  10640