Theorem rexpen 14957

Description: The real numbers are equinumerous to their own Cartesian product, even though it is not necessarily true that ℝ is well-orderable (so we cannot use infxpidm2 8840 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)

Assertion

Ref	Expression
rexpen	⊢ (ℝ × ℝ) ≈ ℝ

Proof of Theorem rexpen

Step	Hyp	Ref	Expression
1		rpnnen 14956	. . . . . 6 ⊢ ℝ ≈ 𝒫 ℕ
2		nnenom 12779	. . . . . . 7 ⊢ ℕ ≈ ω
3		pwen 8133	. . . . . . 7 ⊢ (ℕ ≈ ω → 𝒫 ℕ ≈ 𝒫 ω)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 𝒫 ℕ ≈ 𝒫 ω
5	1, 4	entri 8010	. . . . 5 ⊢ ℝ ≈ 𝒫 ω
6		omex 8540	. . . . . 6 ⊢ ω ∈ V
7	6	pw2en 8067	. . . . 5 ⊢ 𝒫 ω ≈ (2𝑜 ↑𝑚 ω)
8	5, 7	entri 8010	. . . 4 ⊢ ℝ ≈ (2𝑜 ↑𝑚 ω)
9		xpen 8123	. . . 4 ⊢ ((ℝ ≈ (2𝑜 ↑𝑚 ω) ∧ ℝ ≈ (2𝑜 ↑𝑚 ω)) → (ℝ × ℝ) ≈ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)))
10	8, 8, 9	mp2an 708	. . 3 ⊢ (ℝ × ℝ) ≈ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω))
11		2onn 7720	. . . . . . . 8 ⊢ 2𝑜 ∈ ω
12	11	elexi 3213	. . . . . . 7 ⊢ 2𝑜 ∈ V
13	12, 12, 6	xpmapen 8128	. . . . . 6 ⊢ ((2𝑜 × 2𝑜) ↑𝑚 ω) ≈ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω))
14	13	ensymi 8006	. . . . 5 ⊢ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≈ ((2𝑜 × 2𝑜) ↑𝑚 ω)
15		ssid 3624	. . . . . . . . . . . . 13 ⊢ 2𝑜 ⊆ 2𝑜
16		ssnnfi 8179	. . . . . . . . . . . . 13 ⊢ ((2𝑜 ∈ ω ∧ 2𝑜 ⊆ 2𝑜) → 2𝑜 ∈ Fin)
17	11, 15, 16	mp2an 708	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ Fin
18		xpfi 8231	. . . . . . . . . . . 12 ⊢ ((2𝑜 ∈ Fin ∧ 2𝑜 ∈ Fin) → (2𝑜 × 2𝑜) ∈ Fin)
19	17, 17, 18	mp2an 708	. . . . . . . . . . 11 ⊢ (2𝑜 × 2𝑜) ∈ Fin
20		isfinite 8549	. . . . . . . . . . 11 ⊢ ((2𝑜 × 2𝑜) ∈ Fin ↔ (2𝑜 × 2𝑜) ≺ ω)
21	19, 20	mpbi 220	. . . . . . . . . 10 ⊢ (2𝑜 × 2𝑜) ≺ ω
22	6	canth2 8113	. . . . . . . . . 10 ⊢ ω ≺ 𝒫 ω
23		sdomtr 8098	. . . . . . . . . 10 ⊢ (((2𝑜 × 2𝑜) ≺ ω ∧ ω ≺ 𝒫 ω) → (2𝑜 × 2𝑜) ≺ 𝒫 ω)
24	21, 22, 23	mp2an 708	. . . . . . . . 9 ⊢ (2𝑜 × 2𝑜) ≺ 𝒫 ω
25		sdomdom 7983	. . . . . . . . 9 ⊢ ((2𝑜 × 2𝑜) ≺ 𝒫 ω → (2𝑜 × 2𝑜) ≼ 𝒫 ω)
26	24, 25	ax-mp 5	. . . . . . . 8 ⊢ (2𝑜 × 2𝑜) ≼ 𝒫 ω
27		domentr 8015	. . . . . . . 8 ⊢ (((2𝑜 × 2𝑜) ≼ 𝒫 ω ∧ 𝒫 ω ≈ (2𝑜 ↑𝑚 ω)) → (2𝑜 × 2𝑜) ≼ (2𝑜 ↑𝑚 ω))
28	26, 7, 27	mp2an 708	. . . . . . 7 ⊢ (2𝑜 × 2𝑜) ≼ (2𝑜 ↑𝑚 ω)
29		mapdom1 8125	. . . . . . 7 ⊢ ((2𝑜 × 2𝑜) ≼ (2𝑜 ↑𝑚 ω) → ((2𝑜 × 2𝑜) ↑𝑚 ω) ≼ ((2𝑜 ↑𝑚 ω) ↑𝑚 ω))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ ((2𝑜 × 2𝑜) ↑𝑚 ω) ≼ ((2𝑜 ↑𝑚 ω) ↑𝑚 ω)
31		mapxpen 8126	. . . . . . . 8 ⊢ ((2𝑜 ∈ ω ∧ ω ∈ V ∧ ω ∈ V) → ((2𝑜 ↑𝑚 ω) ↑𝑚 ω) ≈ (2𝑜 ↑𝑚 (ω × ω)))
32	11, 6, 6, 31	mp3an 1424	. . . . . . 7 ⊢ ((2𝑜 ↑𝑚 ω) ↑𝑚 ω) ≈ (2𝑜 ↑𝑚 (ω × ω))
33	12	enref 7988	. . . . . . . 8 ⊢ 2𝑜 ≈ 2𝑜
34		xpomen 8838	. . . . . . . 8 ⊢ (ω × ω) ≈ ω
35		mapen 8124	. . . . . . . 8 ⊢ ((2𝑜 ≈ 2𝑜 ∧ (ω × ω) ≈ ω) → (2𝑜 ↑𝑚 (ω × ω)) ≈ (2𝑜 ↑𝑚 ω))
36	33, 34, 35	mp2an 708	. . . . . . 7 ⊢ (2𝑜 ↑𝑚 (ω × ω)) ≈ (2𝑜 ↑𝑚 ω)
37	32, 36	entri 8010	. . . . . 6 ⊢ ((2𝑜 ↑𝑚 ω) ↑𝑚 ω) ≈ (2𝑜 ↑𝑚 ω)
38		domentr 8015	. . . . . 6 ⊢ ((((2𝑜 × 2𝑜) ↑𝑚 ω) ≼ ((2𝑜 ↑𝑚 ω) ↑𝑚 ω) ∧ ((2𝑜 ↑𝑚 ω) ↑𝑚 ω) ≈ (2𝑜 ↑𝑚 ω)) → ((2𝑜 × 2𝑜) ↑𝑚 ω) ≼ (2𝑜 ↑𝑚 ω))
39	30, 37, 38	mp2an 708	. . . . 5 ⊢ ((2𝑜 × 2𝑜) ↑𝑚 ω) ≼ (2𝑜 ↑𝑚 ω)
40		endomtr 8014	. . . . 5 ⊢ ((((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≈ ((2𝑜 × 2𝑜) ↑𝑚 ω) ∧ ((2𝑜 × 2𝑜) ↑𝑚 ω) ≼ (2𝑜 ↑𝑚 ω)) → ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≼ (2𝑜 ↑𝑚 ω))
41	14, 39, 40	mp2an 708	. . . 4 ⊢ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≼ (2𝑜 ↑𝑚 ω)
42		ovex 6678	. . . . . . 7 ⊢ (2𝑜 ↑𝑚 ω) ∈ V
43		0ex 4790	. . . . . . 7 ⊢ ∅ ∈ V
44	42, 43	xpsnen 8044	. . . . . 6 ⊢ ((2𝑜 ↑𝑚 ω) × {∅}) ≈ (2𝑜 ↑𝑚 ω)
45	44	ensymi 8006	. . . . 5 ⊢ (2𝑜 ↑𝑚 ω) ≈ ((2𝑜 ↑𝑚 ω) × {∅})
46		snfi 8038	. . . . . . . . . 10 ⊢ {∅} ∈ Fin
47		isfinite 8549	. . . . . . . . . 10 ⊢ ({∅} ∈ Fin ↔ {∅} ≺ ω)
48	46, 47	mpbi 220	. . . . . . . . 9 ⊢ {∅} ≺ ω
49		sdomtr 8098	. . . . . . . . 9 ⊢ (({∅} ≺ ω ∧ ω ≺ 𝒫 ω) → {∅} ≺ 𝒫 ω)
50	48, 22, 49	mp2an 708	. . . . . . . 8 ⊢ {∅} ≺ 𝒫 ω
51		sdomdom 7983	. . . . . . . 8 ⊢ ({∅} ≺ 𝒫 ω → {∅} ≼ 𝒫 ω)
52	50, 51	ax-mp 5	. . . . . . 7 ⊢ {∅} ≼ 𝒫 ω
53		domentr 8015	. . . . . . 7 ⊢ (({∅} ≼ 𝒫 ω ∧ 𝒫 ω ≈ (2𝑜 ↑𝑚 ω)) → {∅} ≼ (2𝑜 ↑𝑚 ω))
54	52, 7, 53	mp2an 708	. . . . . 6 ⊢ {∅} ≼ (2𝑜 ↑𝑚 ω)
55	42	xpdom2 8055	. . . . . 6 ⊢ ({∅} ≼ (2𝑜 ↑𝑚 ω) → ((2𝑜 ↑𝑚 ω) × {∅}) ≼ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)))
56	54, 55	ax-mp 5	. . . . 5 ⊢ ((2𝑜 ↑𝑚 ω) × {∅}) ≼ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω))
57		endomtr 8014	. . . . 5 ⊢ (((2𝑜 ↑𝑚 ω) ≈ ((2𝑜 ↑𝑚 ω) × {∅}) ∧ ((2𝑜 ↑𝑚 ω) × {∅}) ≼ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω))) → (2𝑜 ↑𝑚 ω) ≼ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)))
58	45, 56, 57	mp2an 708	. . . 4 ⊢ (2𝑜 ↑𝑚 ω) ≼ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω))
59		sbth 8080	. . . 4 ⊢ ((((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≼ (2𝑜 ↑𝑚 ω) ∧ (2𝑜 ↑𝑚 ω) ≼ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω))) → ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≈ (2𝑜 ↑𝑚 ω))
60	41, 58, 59	mp2an 708	. . 3 ⊢ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≈ (2𝑜 ↑𝑚 ω)
61	10, 60	entri 8010	. 2 ⊢ (ℝ × ℝ) ≈ (2𝑜 ↑𝑚 ω)
62	61, 8	entr4i 8013	1 ⊢ (ℝ × ℝ) ≈ ℝ

Syntax hints:   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653   × cxp 5112  (class class class)co 6650  ωcom 7065  2_𝑜c2o 7554   ↑_𝑚 cmap 7857   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  ℝcr 9935  ℕcn 11020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417

This theorem is referenced by:  cpnnen  14958