Theorem pwxpndom2 10722

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 10721	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
2		reldom 8957	. . . . . . 7 ⊢ Rel ≼
3	2	brrelex2i 5704	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		df1o2 8461	. . . . . . . 8 ⊢ 1o = {∅}
5	4	oveq2i 7419	. . . . . . 7 ⊢ (𝐴 ↑m 1o) = (𝐴 ↑m {∅})
6		id 23	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
7		0ex 5260	. . . . . . . . 9 ⊢ ∅ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∅ ∈ V)
9	6, 8	mapsnend 9042	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ↑m {∅}) ≈ 𝐴)
10	5, 9	eqbrtrid 5139	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 1o) ≈ 𝐴)
11		ensym 9008	. . . . . 6 ⊢ ((𝐴 ↑m 1o) ≈ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
12	3, 10, 11	3syl 19	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 ↑m 1o))
13		map2xp 9144	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))
14		ensym 9008	. . . . . 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 8847	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ↑m 1o) → 𝑥:1o⟶𝐴)
17	16	fdmd 6708	. . . . . . . . . 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 8464	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
20	19	sucid 6436	. . . . . . . . . . . 12 ⊢ 1o ∈ suc 1o
21		df-2o 8455	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
22	20, 21	eleqtrri 2859	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
23		1on 8467	. . . . . . . . . . . 12 ⊢ 1o ∈ On
24	23	onirri 6466	. . . . . . . . . . 11 ⊢ ¬ 1o ∈ 1o
25		nelneq2 2885	. . . . . . . . . . 11 ⊢ ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
26	22, 24, 25	mp2an 705	. . . . . . . . . 10 ⊢ ¬ 2o = 1o
27		elmapi 8847	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ↑m 2o) → 𝑥:2o⟶𝐴)
28	27	fdmd 6708	. . . . . . . . . . . 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 2762	. . . . . . . . . 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 3914	. . . . . . . 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 10221	. . . . 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 9622	. . . . . 6 ⊢ ω ∈ V
40		ovex 7441	. . . . . 6 ⊢ (𝐴 ↑m 𝑛) ∈ V
41	39, 40	iunex 7963	. . . . 5 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ V
42		1onn 8627	. . . . . . 7 ⊢ 1o ∈ ω
43		oveq2 7416	. . . . . . . 8 ⊢ (𝑛 = 1o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 1o))
44	43	ssiun2s 5006	. . . . . . 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 8629	. . . . . . 7 ⊢ 2o ∈ ω
47		oveq2 7416	. . . . . . . 8 ⊢ (𝑛 = 2o → (𝐴 ↑m 𝑛) = (𝐴 ↑m 2o))
48	47	ssiun2s 5006	. . . . . . 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 4136	. . . . 5 ⊢ ((𝐴 ↑m 1o) ∪ (𝐴 ↑m 2o)) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)
51		ssdomg 9005	. . . . 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 9017	. . . 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 9012	. . . 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 3450   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  𝒫 cpw 4556  {csn 4583  ∪ ciun 4950   class class class wbr 5102   × cxp 5645  dom cdm 5647  suc csuc 6353  (class class class)co 7408  ωcom 7860  1_oc1o 8447  2_oc2o 8448   ↑_m cmap 8825   ≈ cen 8948   ≼ cdom 8949   ⊔ cdju 9951

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-seqom 8436  df-1o 8454  df-2o 8455  df-oadd 8458  df-omul 8459  df-oexp 8460  df-er 8695  df-map 8827  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-fsupp 9332  df-oi 9482  df-har 9529  df-cnf 9641  df-dju 9954  df-card 9992

This theorem is used by:  pwxpndom  10723  pwdjundom  10724