Theorem pwxpndom2 10262

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 10261	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
2		reldom 8621	. . . . . . 7 ⊢ Rel ≼
3	2	brrelex2i 5595	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		df1o2 8203	. . . . . . . 8 ⊢ 1o = {∅}
5	4	oveq2i 7213	. . . . . . 7 ⊢ (𝐴 ↑m 1o) = (𝐴 ↑m {∅})
6		id 22	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
7		0ex 5189	. . . . . . . . 9 ⊢ ∅ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∅ ∈ V)
9	6, 8	mapsnend 8702	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ↑m {∅}) ≈ 𝐴)
10	5, 9	eqbrtrid 5078	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 1o) ≈ 𝐴)
11		ensym 8666	. . . . . 6 ⊢ ((𝐴 ↑m 1o) ≈ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
12	3, 10, 11	3syl 18	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
13		map2xp 8805	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))
14		ensym 8666	. . . . . 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 8519	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ↑m 1o) → 𝑥:1o⟶𝐴)
17	16	fdmd 6545	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ↑m 1o) → dom 𝑥 = 1o)
18	17	adantr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → dom 𝑥 = 1o)
19		1oex 8204	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
20	19	sucid 6281	. . . . . . . . . . . 12 ⊢ 1o ∈ suc 1o
21		df-2o 8192	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
22	20, 21	eleqtrri 2833	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
23		1on 8198	. . . . . . . . . . . 12 ⊢ 1o ∈ On
24	23	onirri 6309	. . . . . . . . . . 11 ⊢ ¬ 1o ∈ 1o
25		nelneq2 2859	. . . . . . . . . . 11 ⊢ ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
26	22, 24, 25	mp2an 692	. . . . . . . . . 10 ⊢ ¬ 2o = 1o
27		elmapi 8519	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ↑m 2o) → 𝑥:2o⟶𝐴)
28	27	fdmd 6545	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ↑m 2o) → dom 𝑥 = 2o)
29	28	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o)) → dom 𝑥 = 2o)
30	29	eqeq1d 2736	. . . . . . . . . 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 197	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ (𝐴 ↑m 1o) ∧ 𝑥 ∈ (𝐴 ↑m 2o))
33		elin 3873	. . . . . . . 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 4309	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 ↑m 1o) ∩ (𝐴 ↑m 2o)) = ∅)
37		djuenun 9767	. . . . 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 1373	. . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)))
39		omex 9247	. . . . . 6 ⊢ ω ∈ V
40		ovex 7235	. . . . . 6 ⊢ (𝐴 ↑m 𝑛) ∈ V
41	39, 40	iunex 7730	. . . . 5 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ V
42		1onn 8356	. . . . . . 7 ⊢ 1o ∈ ω
43		oveq2 7210	. . . . . . . 8 ⊢ (𝑛 = 1o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 1o))
44	43	ssiun2s 4947	. . . . . . 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 8357	. . . . . . 7 ⊢ 2o ∈ ω
47		oveq2 7210	. . . . . . . 8 ⊢ (𝑛 = 2o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 2o))
48	47	ssiun2s 4947	. . . . . . 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 4089	. . . . 5 ⊢ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)
51		ssdomg 8663	. . . . 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 8675	. . . 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 589	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
55		domtr 8670	. . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) ∧ (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
56	55	expcom 417	. . 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 201	1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  Vcvv 3401   ∪ cun 3855   ∩ cin 3856   ⊆ wss 3857  ∅c0 4227  𝒫 cpw 4503  {csn 4531  ∪ ciun 4894   class class class wbr 5043   × cxp 5538  dom cdm 5540  suc csuc 6204  (class class class)co 7202  ωcom 7633  1_oc1o 8184  2_oc2o 8185   ↑_m cmap 8497   ≈ cen 8612   ≼ cdom 8613   ⊔ cdju 9497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-inf2 9245

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-supp 7893  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-seqom 8173  df-1o 8191  df-2o 8192  df-oadd 8195  df-omul 8196  df-oexp 8197  df-er 8380  df-map 8499  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-fsupp 8975  df-oi 9115  df-har 9162  df-cnf 9266  df-dju 9500  df-card 9538

This theorem is referenced by:  pwxpndom  10263  pwdjundom  10264