Theorem pwxpndom2 9933

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 9932	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
2		reldom 8363	. . . . . . 7 ⊢ Rel ≼
3	2	brrelex2i 5495	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		df1o2 7967	. . . . . . . 8 ⊢ 1o = {∅}
5	4	oveq2i 7027	. . . . . . 7 ⊢ (𝐴 ↑𝑚 1o) = (𝐴 ↑𝑚 {∅})
6		id 22	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
7		0ex 5102	. . . . . . . . 9 ⊢ ∅ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∅ ∈ V)
9	6, 8	mapsnend 8436	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 {∅}) ≈ 𝐴)
10	5, 9	eqbrtrid 4997	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 1o) ≈ 𝐴)
11		ensym 8406	. . . . . 6 ⊢ ((𝐴 ↑𝑚 1o) ≈ 𝐴 → 𝐴 ≈ (𝐴 ↑𝑚 1o))
12	3, 10, 11	3syl 18	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 ↑𝑚 1o))
13		map2xp 8534	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 2o) ≈ (𝐴 × 𝐴))
14		ensym 8406	. . . . . 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 8278	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 1o) → 𝑥:1o⟶𝐴)
17	16	fdmd 6391	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 1o) → dom 𝑥 = 1o)
18	17	adantr 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1o) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2o)) → dom 𝑥 = 1o)
19		1oex 7961	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
20	19	sucid 6145	. . . . . . . . . . . 12 ⊢ 1o ∈ suc 1o
21		df-2o 7954	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
22	20, 21	eleqtrri 2882	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
23		1on 7960	. . . . . . . . . . . 12 ⊢ 1o ∈ On
24	23	onirri 6172	. . . . . . . . . . 11 ⊢ ¬ 1o ∈ 1o
25		nelneq2 2908	. . . . . . . . . . 11 ⊢ ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
26	22, 24, 25	mp2an 688	. . . . . . . . . 10 ⊢ ¬ 2o = 1o
27		elmapi 8278	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 2o) → 𝑥:2o⟶𝐴)
28	27	fdmd 6391	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 2o) → dom 𝑥 = 2o)
29	28	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1o) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2o)) → dom 𝑥 = 2o)
30	29	eqeq1d 2797	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1o) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2o)) → (dom 𝑥 = 1o ↔ 2o = 1o))
31	26, 30	mtbiri 328	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ↑𝑚 1o) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2o)) → ¬ dom 𝑥 = 1o)
32	18, 31	pm2.65i 195	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ (𝐴 ↑𝑚 1o) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2o))
33		elin 4090	. . . . . . . 8 ⊢ (𝑥 ∈ ((𝐴 ↑𝑚 1o) ∩ (𝐴 ↑𝑚 2o)) ↔ (𝑥 ∈ (𝐴 ↑𝑚 1o) ∧ 𝑥 ∈ (𝐴 ↑𝑚 2o)))
34	32, 33	mtbir 324	. . . . . . 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 4277	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 ↑𝑚 1o) ∩ (𝐴 ↑𝑚 2o)) = ∅)
37		djuenun 9442	. . . . 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 1364	. . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴 ↑𝑚 1o) ∪ (𝐴 ↑𝑚 2o)))
39		omex 8952	. . . . . 6 ⊢ ω ∈ V
40		ovex 7048	. . . . . 6 ⊢ (𝐴 ↑𝑚 𝑛) ∈ V
41	39, 40	iunex 7525	. . . . 5 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∈ V
42		1onn 8115	. . . . . . 7 ⊢ 1o ∈ ω
43		oveq2 7024	. . . . . . . 8 ⊢ (𝑛 = 1o → (𝐴 ↑𝑚 𝑛) = (𝐴 ↑𝑚 1o))
44	43	ssiun2s 4871	. . . . . . 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 8116	. . . . . . 7 ⊢ 2o ∈ ω
47		oveq2 7024	. . . . . . . 8 ⊢ (𝑛 = 2o → (𝐴 ↑𝑚 𝑛) = (𝐴 ↑𝑚 2o))
48	47	ssiun2s 4871	. . . . . . 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 4082	. . . . 5 ⊢ ((𝐴 ↑𝑚 1o) ∪ (𝐴 ↑𝑚 2o)) ⊆ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)
51		ssdomg 8403	. . . . 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 8415	. . . 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 586	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
55		domtr 8410	. . . 4 ⊢ ((𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) ∧ (𝐴 ⊔ (𝐴 × 𝐴)) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)) → 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
56	55	expcom 414	. . 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 199	1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ∪ cun 3857   ∩ cin 3858   ⊆ wss 3859  ∅c0 4211  𝒫 cpw 4453  {csn 4472  ∪ ciun 4825   class class class wbr 4962   × cxp 5441  dom cdm 5443  suc csuc 6068  (class class class)co 7016  ωcom 7436  1_oc1o 7946  2_oc2o 7947   ↑_𝑚 cmap 8256   ≈ cen 8354   ≼ cdom 8355   ⊔ cdju 9173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-seqom 7935  df-1o 7953  df-2o 7954  df-oadd 7957  df-omul 7958  df-oexp 7959  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-oi 8820  df-har 8868  df-cnf 8971  df-dju 9176  df-card 9214

This theorem is referenced by:  pwxpndom  9934  pwdjundom  9935