Theorem canthp1 10076

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 ⊔ 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 8717	. . . . 5 ⊢ 1o ≺ 2o
2		sdomdom 8537	. . . . 5 ⊢ (1o ≺ 2o → 1o ≼ 2o)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1o ≼ 2o
4		relsdom 8516	. . . . 5 ⊢ Rel ≺
5	4	brrelex2i 5609	. . . 4 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
6		djudom2 9609	. . . 4 ⊢ ((1o ≼ 2o ∧ 𝐴 ∈ V) → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
7	3, 5, 6	sylancr 589	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
8		canthp1lem1 10074	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 2o) ≼ 𝒫 𝐴)
9		domtr 8562	. . 3 ⊢ (((𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o) ∧ (𝐴 ⊔ 2o) ≼ 𝒫 𝐴) → (𝐴 ⊔ 1o) ≼ 𝒫 𝐴)
10	7, 8, 9	syl2anc 586	. 2 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≼ 𝒫 𝐴)
11		fal 1551	. . 3 ⊢ ¬ ⊥
12		ensym 8558	. . . . 5 ⊢ ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → 𝒫 𝐴 ≈ (𝐴 ⊔ 1o))
13		bren 8518	. . . . 5 ⊢ (𝒫 𝐴 ≈ (𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
14	12, 13	sylib 220	. . . 4 ⊢ ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
15		f1of 6615	. . . . . . . . . 10 ⊢ (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → 𝑓:𝒫 𝐴⟶(𝐴 ⊔ 1o))
16		pwidg 4561	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
17	5, 16	syl 17	. . . . . . . . . 10 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ 𝒫 𝐴)
18		ffvelrn 6849	. . . . . . . . . 10 ⊢ ((𝑓:𝒫 𝐴⟶(𝐴 ⊔ 1o) ∧ 𝐴 ∈ 𝒫 𝐴) → (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o))
19	15, 17, 18	syl2anr 598	. . . . . . . . 9 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o))
20		dju1dif 9598	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
21	5, 19, 20	syl2an2r 683	. . . . . . . 8 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
22		bren 8518	. . . . . . . 8 ⊢ (((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴 ↔ ∃𝑔 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
23	21, 22	sylib 220	. . . . . . 7 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ∃𝑔 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
24		simpll 765	. . . . . . . . 9 ⊢ (((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 1o ≺ 𝐴)
25		simplr 767	. . . . . . . . 9 ⊢ (((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
26		simpr 487	. . . . . . . . 9 ⊢ (((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
27		eqeq1 2825	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝐴 ↔ 𝑥 = 𝐴))
28		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥)
29	27, 28	ifbieq2d 4492	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → if(𝑤 = 𝐴, ∅, 𝑤) = if(𝑥 = 𝐴, ∅, 𝑥))
30	29	cbvmptv 5169	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)) = (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))
31	30	coeq2i 5731	. . . . . . . . 9 ⊢ ((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤))) = ((𝑔 ∘ 𝑓) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
32		eqid 2821	. . . . . . . . . 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))}
33	32	fpwwecbv 10066	. . . . . . . . 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))}
34		eqid 2821	. . . . . . . . 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))}
35	24, 25, 26, 31, 33, 34	canthp1lem2 10075	. . . . . . . 8 ⊢ ¬ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
36	35	pm2.21i 119	. . . . . . 7 ⊢ (((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → ⊥)
37	23, 36	exlimddv 1936	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ⊥)
38	37	ex 415	. . . . 5 ⊢ (1o ≺ 𝐴 → (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → ⊥))
39	38	exlimdv 1934	. . . 4 ⊢ (1o ≺ 𝐴 → (∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → ⊥))
40	14, 39	syl5 34	. . 3 ⊢ (1o ≺ 𝐴 → ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → ⊥))
41	11, 40	mtoi 201	. 2 ⊢ (1o ≺ 𝐴 → ¬ (𝐴 ⊔ 1o) ≈ 𝒫 𝐴)
42		brsdom 8532	. 2 ⊢ ((𝐴 ⊔ 1o) ≺ 𝒫 𝐴 ↔ ((𝐴 ⊔ 1o) ≼ 𝒫 𝐴 ∧ ¬ (𝐴 ⊔ 1o) ≈ 𝒫 𝐴))
43	10, 41, 42	sylanbrc 585	1 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≺ 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537  ⊥wfal 1549  ∃wex 1780   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291  ifcif 4467  𝒫 cpw 4539  {csn 4567  ∪ cuni 4838   class class class wbr 5066  {copab 5128   ↦ cmpt 5146   We wwe 5513   × cxp 5553  ^◡ccnv 5554  dom cdm 5555   “ cima 5558   ∘ ccom 5559  ⟶wf 6351  –1-1-onto→wf1o 6354  ‘cfv 6355  1_oc1o 8095  2_oc2o 8096   ≈ cen 8506   ≼ cdom 8507   ≺ csdm 8508   ⊔ cdju 9327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-oi 8974  df-dju 9330  df-card 9368

This theorem is referenced by:  finngch  10077  gchdju1  10078