Theorem canthp1 9476

Description: A slightly stronger form of Cantor's theorem: For 1 < 𝑛, 𝑛 + 1 < 2↑𝑛. Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
canthp1	⊢ (1𝑜 ≺ 𝐴 → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)

Proof of Theorem canthp1

Dummy variables 𝑓 𝑎 𝑔 𝑟 𝑠 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1sdom2 8159	. . . 4 ⊢ 1𝑜 ≺ 2𝑜
2		sdomdom 7983	. . . 4 ⊢ (1𝑜 ≺ 2𝑜 → 1𝑜 ≼ 2𝑜)
3		cdadom2 9009	. . . 4 ⊢ (1𝑜 ≼ 2𝑜 → (𝐴 +𝑐 1𝑜) ≼ (𝐴 +𝑐 2𝑜))
4	1, 2, 3	mp2b 10	. . 3 ⊢ (𝐴 +𝑐 1𝑜) ≼ (𝐴 +𝑐 2𝑜)
5		canthp1lem1 9474	. . 3 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)
6		domtr 8009	. . 3 ⊢ (((𝐴 +𝑐 1𝑜) ≼ (𝐴 +𝑐 2𝑜) ∧ (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴) → (𝐴 +𝑐 1𝑜) ≼ 𝒫 𝐴)
7	4, 5, 6	sylancr 695	. 2 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 +𝑐 1𝑜) ≼ 𝒫 𝐴)
8		fal 1490	. . 3 ⊢ ¬ ⊥
9		ensym 8005	. . . . 5 ⊢ ((𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴 → 𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜))
10		bren 7964	. . . . 5 ⊢ (𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜) ↔ ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜))
11	9, 10	sylib 208	. . . 4 ⊢ ((𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴 → ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜))
12		f1of 6137	. . . . . . . . . 10 ⊢ (𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜) → 𝑓:𝒫 𝐴⟶(𝐴 +𝑐 1𝑜))
13		relsdom 7962	. . . . . . . . . . . 12 ⊢ Rel ≺
14	13	brrelex2i 5159	. . . . . . . . . . 11 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ V)
15		pwidg 4173	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
16	14, 15	syl 17	. . . . . . . . . 10 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ 𝒫 𝐴)
17		ffvelrn 6357	. . . . . . . . . 10 ⊢ ((𝑓:𝒫 𝐴⟶(𝐴 +𝑐 1𝑜) ∧ 𝐴 ∈ 𝒫 𝐴) → (𝑓‘𝐴) ∈ (𝐴 +𝑐 1𝑜))
18	12, 16, 17	syl2anr 495	. . . . . . . . 9 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) → (𝑓‘𝐴) ∈ (𝐴 +𝑐 1𝑜))
19		cda1dif 8998	. . . . . . . . 9 ⊢ ((𝑓‘𝐴) ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
20	18, 19	syl 17	. . . . . . . 8 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) → ((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
21		bren 7964	. . . . . . . 8 ⊢ (((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)}) ≈ 𝐴 ↔ ∃𝑔 𝑔:((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
22	20, 21	sylib 208	. . . . . . 7 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) → ∃𝑔 𝑔:((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
23		simpll 790	. . . . . . . . 9 ⊢ (((1𝑜 ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) ∧ 𝑔:((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 1𝑜 ≺ 𝐴)
24		simplr 792	. . . . . . . . 9 ⊢ (((1𝑜 ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) ∧ 𝑔:((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜))
25		simpr 477	. . . . . . . . 9 ⊢ (((1𝑜 ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) ∧ 𝑔:((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 𝑔:((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
26		eqeq1 2626	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝐴 ↔ 𝑥 = 𝐴))
27		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥)
28	26, 27	ifbieq2d 4111	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → if(𝑤 = 𝐴, ∅, 𝑤) = if(𝑥 = 𝐴, ∅, 𝑥))
29	28	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)) = (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))
30	29	coeq2i 5282	. . . . . . . . 9 ⊢ ((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤))) = ((𝑔 ∘ 𝑓) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
31		eqid 2622	. . . . . . . . . 10 ⊢ {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)))‘(◡𝑠 “ {𝑧})) = 𝑧))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)))‘(◡𝑠 “ {𝑧})) = 𝑧))}
32	31	fpwwecbv 9466	. . . . . . . . 9 ⊢ {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)))‘(◡𝑠 “ {𝑧})) = 𝑧))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)))‘(◡𝑟 “ {𝑦})) = 𝑦))}
33		eqid 2622	. . . . . . . . 9 ⊢ ∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)))‘(◡𝑠 “ {𝑧})) = 𝑧))} = ∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)))‘(◡𝑠 “ {𝑧})) = 𝑧))}
34	23, 24, 25, 30, 32, 33	canthp1lem2 9475	. . . . . . . 8 ⊢ ¬ ((1𝑜 ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) ∧ 𝑔:((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
35	34	pm2.21i 116	. . . . . . 7 ⊢ (((1𝑜 ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) ∧ 𝑔:((𝐴 +𝑐 1𝑜) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → ⊥)
36	22, 35	exlimddv 1863	. . . . . 6 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) → ⊥)
37	36	ex 450	. . . . 5 ⊢ (1𝑜 ≺ 𝐴 → (𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜) → ⊥))
38	37	exlimdv 1861	. . . 4 ⊢ (1𝑜 ≺ 𝐴 → (∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜) → ⊥))
39	11, 38	syl5 34	. . 3 ⊢ (1𝑜 ≺ 𝐴 → ((𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴 → ⊥))
40	8, 39	mtoi 190	. 2 ⊢ (1𝑜 ≺ 𝐴 → ¬ (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴)
41		brsdom 7978	. 2 ⊢ ((𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴 ↔ ((𝐴 +𝑐 1𝑜) ≼ 𝒫 𝐴 ∧ ¬ (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴))
42	7, 40, 41	sylanbrc 698	1 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483  ⊥wfal 1488  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   We wwe 5072   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   “ cima 5117   ∘ ccom 5118  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954   +_𝑐 ccda 8989

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

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-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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990

This theorem is referenced by:  finngch  9477  gchcda1  9478