Theorem rexpen 14796

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 8723 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 14795	. . . . . 6 ⊢ ℝ ≈ 𝒫 ℕ
2		nnenom 12641	. . . . . . 7 ⊢ ℕ ≈ ω
3		pwen 8018	. . . . . . 7 ⊢ (ℕ ≈ ω → 𝒫 ℕ ≈ 𝒫 ω)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 𝒫 ℕ ≈ 𝒫 ω
5	1, 4	entri 7896	. . . . 5 ⊢ ℝ ≈ 𝒫 ω
6		omex 8423	. . . . . 6 ⊢ ω ∈ V
7	6	pw2en 7952	. . . . 5 ⊢ 𝒫 ω ≈ (2𝑜 ↑𝑚 ω)
8	5, 7	entri 7896	. . . 4 ⊢ ℝ ≈ (2𝑜 ↑𝑚 ω)
9		xpen 8008	. . . 4 ⊢ ((ℝ ≈ (2𝑜 ↑𝑚 ω) ∧ ℝ ≈ (2𝑜 ↑𝑚 ω)) → (ℝ × ℝ) ≈ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)))
10	8, 8, 9	mp2an 704	. . 3 ⊢ (ℝ × ℝ) ≈ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω))
11		2onn 7607	. . . . . . . 8 ⊢ 2𝑜 ∈ ω
12	11	elexi 3186	. . . . . . 7 ⊢ 2𝑜 ∈ V
13	12, 12, 6	xpmapen 8013	. . . . . 6 ⊢ ((2𝑜 × 2𝑜) ↑𝑚 ω) ≈ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω))
14	13	ensymi 7892	. . . . 5 ⊢ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≈ ((2𝑜 × 2𝑜) ↑𝑚 ω)
15		ssid 3587	. . . . . . . . . . . . 13 ⊢ 2𝑜 ⊆ 2𝑜
16		ssnnfi 8064	. . . . . . . . . . . . 13 ⊢ ((2𝑜 ∈ ω ∧ 2𝑜 ⊆ 2𝑜) → 2𝑜 ∈ Fin)
17	11, 15, 16	mp2an 704	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ Fin
18		xpfi 8116	. . . . . . . . . . . 12 ⊢ ((2𝑜 ∈ Fin ∧ 2𝑜 ∈ Fin) → (2𝑜 × 2𝑜) ∈ Fin)
19	17, 17, 18	mp2an 704	. . . . . . . . . . 11 ⊢ (2𝑜 × 2𝑜) ∈ Fin
20		isfinite 8432	. . . . . . . . . . 11 ⊢ ((2𝑜 × 2𝑜) ∈ Fin ↔ (2𝑜 × 2𝑜) ≺ ω)
21	19, 20	mpbi 219	. . . . . . . . . 10 ⊢ (2𝑜 × 2𝑜) ≺ ω
22	6	canth2 7998	. . . . . . . . . 10 ⊢ ω ≺ 𝒫 ω
23		sdomtr 7983	. . . . . . . . . 10 ⊢ (((2𝑜 × 2𝑜) ≺ ω ∧ ω ≺ 𝒫 ω) → (2𝑜 × 2𝑜) ≺ 𝒫 ω)
24	21, 22, 23	mp2an 704	. . . . . . . . 9 ⊢ (2𝑜 × 2𝑜) ≺ 𝒫 ω
25		sdomdom 7869	. . . . . . . . 9 ⊢ ((2𝑜 × 2𝑜) ≺ 𝒫 ω → (2𝑜 × 2𝑜) ≼ 𝒫 ω)
26	24, 25	ax-mp 5	. . . . . . . 8 ⊢ (2𝑜 × 2𝑜) ≼ 𝒫 ω
27		domentr 7901	. . . . . . . 8 ⊢ (((2𝑜 × 2𝑜) ≼ 𝒫 ω ∧ 𝒫 ω ≈ (2𝑜 ↑𝑚 ω)) → (2𝑜 × 2𝑜) ≼ (2𝑜 ↑𝑚 ω))
28	26, 7, 27	mp2an 704	. . . . . . 7 ⊢ (2𝑜 × 2𝑜) ≼ (2𝑜 ↑𝑚 ω)
29		mapdom1 8010	. . . . . . 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 8011	. . . . . . . 8 ⊢ ((2𝑜 ∈ ω ∧ ω ∈ V ∧ ω ∈ V) → ((2𝑜 ↑𝑚 ω) ↑𝑚 ω) ≈ (2𝑜 ↑𝑚 (ω × ω)))
32	11, 6, 6, 31	mp3an 1416	. . . . . . 7 ⊢ ((2𝑜 ↑𝑚 ω) ↑𝑚 ω) ≈ (2𝑜 ↑𝑚 (ω × ω))
33	12	enref 7874	. . . . . . . 8 ⊢ 2𝑜 ≈ 2𝑜
34		xpomen 8721	. . . . . . . 8 ⊢ (ω × ω) ≈ ω
35		mapen 8009	. . . . . . . 8 ⊢ ((2𝑜 ≈ 2𝑜 ∧ (ω × ω) ≈ ω) → (2𝑜 ↑𝑚 (ω × ω)) ≈ (2𝑜 ↑𝑚 ω))
36	33, 34, 35	mp2an 704	. . . . . . 7 ⊢ (2𝑜 ↑𝑚 (ω × ω)) ≈ (2𝑜 ↑𝑚 ω)
37	32, 36	entri 7896	. . . . . 6 ⊢ ((2𝑜 ↑𝑚 ω) ↑𝑚 ω) ≈ (2𝑜 ↑𝑚 ω)
38		domentr 7901	. . . . . 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 704	. . . . 5 ⊢ ((2𝑜 × 2𝑜) ↑𝑚 ω) ≼ (2𝑜 ↑𝑚 ω)
40		endomtr 7900	. . . . 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 704	. . . 4 ⊢ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≼ (2𝑜 ↑𝑚 ω)
42		ovex 6577	. . . . . . 7 ⊢ (2𝑜 ↑𝑚 ω) ∈ V
43		0ex 4718	. . . . . . 7 ⊢ ∅ ∈ V
44	42, 43	xpsnen 7929	. . . . . 6 ⊢ ((2𝑜 ↑𝑚 ω) × {∅}) ≈ (2𝑜 ↑𝑚 ω)
45	44	ensymi 7892	. . . . 5 ⊢ (2𝑜 ↑𝑚 ω) ≈ ((2𝑜 ↑𝑚 ω) × {∅})
46		snfi 7923	. . . . . . . . . 10 ⊢ {∅} ∈ Fin
47		isfinite 8432	. . . . . . . . . 10 ⊢ ({∅} ∈ Fin ↔ {∅} ≺ ω)
48	46, 47	mpbi 219	. . . . . . . . 9 ⊢ {∅} ≺ ω
49		sdomtr 7983	. . . . . . . . 9 ⊢ (({∅} ≺ ω ∧ ω ≺ 𝒫 ω) → {∅} ≺ 𝒫 ω)
50	48, 22, 49	mp2an 704	. . . . . . . 8 ⊢ {∅} ≺ 𝒫 ω
51		sdomdom 7869	. . . . . . . 8 ⊢ ({∅} ≺ 𝒫 ω → {∅} ≼ 𝒫 ω)
52	50, 51	ax-mp 5	. . . . . . 7 ⊢ {∅} ≼ 𝒫 ω
53		domentr 7901	. . . . . . 7 ⊢ (({∅} ≼ 𝒫 ω ∧ 𝒫 ω ≈ (2𝑜 ↑𝑚 ω)) → {∅} ≼ (2𝑜 ↑𝑚 ω))
54	52, 7, 53	mp2an 704	. . . . . 6 ⊢ {∅} ≼ (2𝑜 ↑𝑚 ω)
55	42	xpdom2 7940	. . . . . 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 7900	. . . . 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 704	. . . 4 ⊢ (2𝑜 ↑𝑚 ω) ≼ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω))
59		sbth 7965	. . . 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 704	. . 3 ⊢ ((2𝑜 ↑𝑚 ω) × (2𝑜 ↑𝑚 ω)) ≈ (2𝑜 ↑𝑚 ω)
61	10, 60	entri 7896	. 2 ⊢ (ℝ × ℝ) ≈ (2𝑜 ↑𝑚 ω)
62	61, 8	entr4i 7899	1 ⊢ (ℝ × ℝ) ≈ ℝ

Syntax hints:   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583   × cxp 5036  (class class class)co 6549  ωcom 6957  2_𝑜c2o 7441   ↑_𝑚 cmap 7744   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841  ℝcr 9814  ℕcn 10897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265

This theorem is referenced by:  cpnnen  14797