Theorem pwxpndom2 10677

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 10676	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
2		reldom 8961	. . . . . . 7 ⊢ Rel ≼
3	2	brrelex2i 5716	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		df1o2 8465	. . . . . . . 8 ⊢ 1o = {∅}
5	4	oveq2i 7427	. . . . . . 7 ⊢ (𝐴 ↑m 1o) = (𝐴 ↑m {∅})
6		id 23	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
7		0ex 5268	. . . . . . . . 9 ⊢ ∅ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∅ ∈ V)
9	6, 8	mapsnend 9046	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ↑m {∅}) ≈ 𝐴)
10	5, 9	eqbrtrid 5144	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 1o) ≈ 𝐴)
11		ensym 9012	. . . . . 6 ⊢ ((𝐴 ↑m 1o) ≈ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
12	3, 10, 11	3syl 19	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
13		map2xp 9148	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))
14		ensym 9012	. . . . . 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 8851	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ↑m 1o) → 𝑥:1o⟶𝐴)
17	16	fdmd 6717	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ↑m 1o) → dom 𝑥 = 1o)
18	17	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → dom 𝑥 = 1o)
19		1oex 8468	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
20	19	sucid 6446	. . . . . . . . . . . 12 ⊢ 1o ∈ suc 1o
21		df-2o 8459	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
22	20, 21	eleqtrri 2861	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
23		1on 8471	. . . . . . . . . . . 12 ⊢ 1o ∈ On
24	23	onirri 6476	. . . . . . . . . . 11 ⊢ ¬ 1o ∈ 1o
25		nelneq2 2887	. . . . . . . . . . 11 ⊢ ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
26	22, 24, 25	mp2an 705	. . . . . . . . . 10 ⊢ ¬ 2o = 1o
27		elmapi 8851	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ↑m 2o) → 𝑥:2o⟶𝐴)
28	27	fdmd 6717	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ↑m 2o) → dom 𝑥 = 2o)
29	28	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → dom 𝑥 = 2o)
30	29	eqeq1d 2764	. . . . . . . . . 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 3918	. . . . . . . 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 4368	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 ↑m 1o) ∩ (𝐴 ↑m 2o)) = ∅)
37		djuenun 10176	. . . . 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 1398	. . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)))
39		omex 9625	. . . . . 6 ⊢ ω ∈ V
40		ovex 7449	. . . . . 6 ⊢ (𝐴 ↑m 𝑛) ∈ V
41	39, 40	iunex 7968	. . . . 5 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ V
42		1onn 8631	. . . . . . 7 ⊢ 1o ∈ ω
43		oveq2 7424	. . . . . . . 8 ⊢ (𝑛 = 1o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 1o))
44	43	ssiun2s 5011	. . . . . . 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 8633	. . . . . . 7 ⊢ 2o ∈ ω
47		oveq2 7424	. . . . . . . 8 ⊢ (𝑛 = 2o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 2o))
48	47	ssiun2s 5011	. . . . . . 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 4140	. . . . 5 ⊢ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)
51		ssdomg 9009	. . . . 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 9021	. . . 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 598	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
55		domtr 9016	. . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) ∧ (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
56	55	expcom 419	. . 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560  {csn 4587  ∪ ciun 4954   class class class wbr 5107   × cxp 5657  dom cdm 5659  suc csuc 6363  (class class class)co 7416  ωcom 7865  1_oc1o 8451  2_oc2o 8452   ↑_m cmap 8829   ≈ cen 8952   ≼ cdom 8953   ⊔ cdju 9906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-seqom 8440  df-1o 8458  df-2o 8459  df-oadd 8462  df-omul 8463  df-oexp 8464  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-oi 9485  df-har 9532  df-cnf 9644  df-dju 9909  df-card 9947

This theorem is used by:  pwxpndom  10678  pwdjundom  10679