Theorem canthp1 10649

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 9240	. . . . 5 ⊢ 1o ≺ 2o
2		sdomdom 8976	. . . . 5 ⊢ (1o ≺ 2o → 1o ≼ 2o)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1o ≼ 2o
4		relsdom 8946	. . . . 5 ⊢ Rel ≺
5	4	brrelex2i 5734	. . . 4 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
6		djudom2 10178	. . . 4 ⊢ ((1o ≼ 2o ∧ 𝐴 ∈ V) → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
7	3, 5, 6	sylancr 588	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
8		canthp1lem1 10647	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 2o) ≼ 𝒫 𝐴)
9		domtr 9003	. . 3 ⊢ (((𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o) ∧ (𝐴 ⊔ 2o) ≼ 𝒫 𝐴) → (𝐴 ⊔ 1o) ≼ 𝒫 𝐴)
10	7, 8, 9	syl2anc 585	. 2 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≼ 𝒫 𝐴)
11		fal 1556	. . 3 ⊢ ¬ ⊥
12		ensym 8999	. . . . 5 ⊢ ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → 𝒫 𝐴 ≈ (𝐴 ⊔ 1o))
13		bren 8949	. . . . 5 ⊢ (𝒫 𝐴 ≈ (𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
14	12, 13	sylib 217	. . . 4 ⊢ ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
15		f1of 6834	. . . . . . . . . 10 ⊢ (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → 𝑓:𝒫 𝐴⟶(𝐴 ⊔ 1o))
16		pwidg 4623	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
17	5, 16	syl 17	. . . . . . . . . 10 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ 𝒫 𝐴)
18		ffvelcdm 7084	. . . . . . . . . 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 10167	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
21	5, 19, 20	syl2an2r 684	. . . . . . . 8 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
22		bren 8949	. . . . . . . 8 ⊢ (((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴 ↔ ∃𝑔 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
23	21, 22	sylib 217	. . . . . . 7 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ∃𝑔 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
24		simpll 766	. . . . . . . . 9 ⊢ (((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 1o ≺ 𝐴)
25		simplr 768	. . . . . . . . 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 486	. . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝐴 ↔ 𝑥 = 𝐴))
28		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥)
29	27, 28	ifbieq2d 4555	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → if(𝑤 = 𝐴, ∅, 𝑤) = if(𝑥 = 𝐴, ∅, 𝑥))
30	29	cbvmptv 5262	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)) = (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))
31	30	coeq2i 5861	. . . . . . . . 9 ⊢ ((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤))) = ((𝑔 ∘ 𝑓) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
32		eqid 2733	. . . . . . . . . 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 10639	. . . . . . . . 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 2733	. . . . . . . . 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 10648	. . . . . . . 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 1939	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ⊥)
38	37	ex 414	. . . . 5 ⊢ (1o ≺ 𝐴 → (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → ⊥))
39	38	exlimdv 1937	. . . 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 198	. 2 ⊢ (1o ≺ 𝐴 → ¬ (𝐴 ⊔ 1o) ≈ 𝒫 𝐴)
42		brsdom 8971	. 2 ⊢ ((𝐴 ⊔ 1o) ≺ 𝒫 𝐴 ↔ ((𝐴 ⊔ 1o) ≼ 𝒫 𝐴 ∧ ¬ (𝐴 ⊔ 1o) ≈ 𝒫 𝐴))
43	10, 41, 42	sylanbrc 584	1 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≺ 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542  ⊥wfal 1554  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4323  ifcif 4529  𝒫 cpw 4603  {csn 4629  ∪ cuni 4909   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   We wwe 5631   × cxp 5675  ^◡ccnv 5676  dom cdm 5677   “ cima 5680   ∘ ccom 5681  ⟶wf 6540  –1-1-onto→wf1o 6543  ‘cfv 6544  1_oc1o 8459  2_oc2o 8460   ≈ cen 8936   ≼ cdom 8937   ≺ csdm 8938   ⊔ cdju 9893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-dju 9896  df-card 9934

This theorem is referenced by:  finngch  10650  gchdju1  10651