Theorem canthp1 10577

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 9160	. . . . 5 ⊢ 1o ≺ 2o
2		sdomdom 8929	. . . . 5 ⊢ (1o ≺ 2o → 1o ≼ 2o)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1o ≼ 2o
4		relsdom 8902	. . . . 5 ⊢ Rel ≺
5	4	brrelex2i 5689	. . . 4 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
6		djudom2 10106	. . . 4 ⊢ ((1o ≼ 2o ∧ 𝐴 ∈ V) → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
7	3, 5, 6	sylancr 588	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
8		canthp1lem1 10575	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 2o) ≼ 𝒫 𝐴)
9		domtr 8956	. . 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 8952	. . . . 5 ⊢ ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → 𝒫 𝐴 ≈ (𝐴 ⊔ 1o))
13		bren 8905	. . . . 5 ⊢ (𝒫 𝐴 ≈ (𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
14	12, 13	sylib 218	. . . 4 ⊢ ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
15		f1of 6782	. . . . . . . . . 10 ⊢ (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → 𝑓:𝒫 𝐴⟶(𝐴 ⊔ 1o))
16		pwidg 4576	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
17	5, 16	syl 17	. . . . . . . . . 10 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ 𝒫 𝐴)
18		ffvelcdm 7035	. . . . . . . . . 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 10095	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
21	5, 19, 20	syl2an2r 686	. . . . . . . 8 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
22		bren 8905	. . . . . . . 8 ⊢ (((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴 ↔ ∃𝑔 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
23	21, 22	sylib 218	. . . . . . 7 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ∃𝑔 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
24		simpll 767	. . . . . . . . 9 ⊢ (((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 1o ≺ 𝐴)
25		simplr 769	. . . . . . . . 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 484	. . . . . . . . 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 2741	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝐴 ↔ 𝑥 = 𝐴))
28		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥)
29	27, 28	ifbieq2d 4508	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → if(𝑤 = 𝐴, ∅, 𝑤) = if(𝑥 = 𝐴, ∅, 𝑥))
30	29	cbvmptv 5204	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)) = (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))
31	30	coeq2i 5817	. . . . . . . . 9 ⊢ ((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤))) = ((𝑔 ∘ 𝑓) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
32		eqid 2737	. . . . . . . . . 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 10567	. . . . . . . . 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 2737	. . . . . . . . 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 10576	. . . . . . . 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 1937	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ⊥)
38	37	ex 412	. . . . 5 ⊢ (1o ≺ 𝐴 → (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → ⊥))
39	38	exlimdv 1935	. . . 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 199	. 2 ⊢ (1o ≺ 𝐴 → ¬ (𝐴 ⊔ 1o) ≈ 𝒫 𝐴)
42		brsdom 8923	. 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 395   = wceq 1542  ⊥wfal 1554  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  ifcif 4481  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865   class class class wbr 5100  {copab 5162   ↦ cmpt 5181   We wwe 5584   × cxp 5630  ^◡ccnv 5631  dom cdm 5632   “ cima 5635   ∘ ccom 5636  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  1_oc1o 8400  2_oc2o 8401   ≈ cen 8892   ≼ cdom 8893   ≺ csdm 8894   ⊔ cdju 9822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-oi 9427  df-dju 9825  df-card 9863

This theorem is referenced by:  finngch  10578  gchdju1  10579