Theorem pwxpndom2 10662

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 10661	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
2		reldom 8947	. . . . . . 7 ⊢ Rel ≼
3	2	brrelex2i 5732	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		df1o2 8475	. . . . . . . 8 ⊢ 1o = {∅}
5	4	oveq2i 7422	. . . . . . 7 ⊢ (𝐴 ↑m 1o) = (𝐴 ↑m {∅})
6		id 22	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
7		0ex 5306	. . . . . . . . 9 ⊢ ∅ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∅ ∈ V)
9	6, 8	mapsnend 9038	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ↑m {∅}) ≈ 𝐴)
10	5, 9	eqbrtrid 5182	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 1o) ≈ 𝐴)
11		ensym 9001	. . . . . 6 ⊢ ((𝐴 ↑m 1o) ≈ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
12	3, 10, 11	3syl 18	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
13		map2xp 9149	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))
14		ensym 9001	. . . . . 6 ⊢ ((𝐴 ↑m 2o) ≈ (𝐴 × 𝐴) → (𝐴 × 𝐴) ≈ (𝐴 ↑m 2o))
15	3, 13, 14	3syl 18	. . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ (𝐴 ↑m 2o))
16		elmapi 8845	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ↑m 1o) → 𝑥:1o⟶𝐴)
17	16	fdmd 6727	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ↑m 1o) → dom 𝑥 = 1o)
18	17	adantr 479	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → dom 𝑥 = 1o)
19		1oex 8478	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
20	19	sucid 6445	. . . . . . . . . . . 12 ⊢ 1o ∈ suc 1o
21		df-2o 8469	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
22	20, 21	eleqtrri 2830	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
23		1on 8480	. . . . . . . . . . . 12 ⊢ 1o ∈ On
24	23	onirri 6476	. . . . . . . . . . 11 ⊢ ¬ 1o ∈ 1o
25		nelneq2 2856	. . . . . . . . . . 11 ⊢ ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
26	22, 24, 25	mp2an 688	. . . . . . . . . 10 ⊢ ¬ 2o = 1o
27		elmapi 8845	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ↑m 2o) → 𝑥:2o⟶𝐴)
28	27	fdmd 6727	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ↑m 2o) → dom 𝑥 = 2o)
29	28	adantl 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → dom 𝑥 = 2o)
30	29	eqeq1d 2732	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → (dom 𝑥 = 1o ↔ 2o = 1o))
31	26, 30	mtbiri 326	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → ¬ dom 𝑥 = 1o)
32	18, 31	pm2.65i 193	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o))
33		elin 3963	. . . . . . . 8 ⊢ (𝑥 ∈ ((𝐴 ↑m 1o) ∩ (𝐴 ↑m 2o)) ↔ (𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)))
34	32, 33	mtbir 322	. . . . . . 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 4403	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 ↑m 1o) ∩ (𝐴 ↑m 2o)) = ∅)
37		djuenun 10167	. . . . 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 1369	. . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)))
39		omex 9640	. . . . . 6 ⊢ ω ∈ V
40		ovex 7444	. . . . . 6 ⊢ (𝐴 ↑m 𝑛) ∈ V
41	39, 40	iunex 7957	. . . . 5 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ V
42		1onn 8641	. . . . . . 7 ⊢ 1o ∈ ω
43		oveq2 7419	. . . . . . . 8 ⊢ (𝑛 = 1o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 1o))
44	43	ssiun2s 5050	. . . . . . 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 8643	. . . . . . 7 ⊢ 2o ∈ ω
47		oveq2 7419	. . . . . . . 8 ⊢ (𝑛 = 2o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 2o))
48	47	ssiun2s 5050	. . . . . . 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 4184	. . . . 5 ⊢ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)
51		ssdomg 8998	. . . . 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 9010	. . . 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 584	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
55		domtr 9005	. . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) ∧ (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
56	55	expcom 412	. . 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 17	. 2 ⊢ (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
58	1, 57	mtod 197	1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  {csn 4627  ∪ ciun 4996   class class class wbr 5147   × cxp 5673  dom cdm 5675  suc csuc 6365  (class class class)co 7411  ωcom 7857  1_oc1o 8461  2_oc2o 8462   ↑_m cmap 8822   ≈ cen 8938   ≼ cdom 8939   ⊔ cdju 9895

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-seqom 8450  df-1o 8468  df-2o 8469  df-oadd 8472  df-omul 8473  df-oexp 8474  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-har 9554  df-cnf 9659  df-dju 9898  df-card 9936

This theorem is referenced by:  pwxpndom  10663  pwdjundom  10664