Theorem canthp1 10634

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 9204	. . . . 5 ⊢ 1o ≺ 2o
2		sdomdom 8973	. . . . 5 ⊢ (1o ≺ 2o → 1o ≼ 2o)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1o ≼ 2o
4		relsdom 8946	. . . . 5 ⊢ Rel ≺
5	4	brrelex2i 5718	. . . 4 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
6		djudom2 10163	. . . 4 ⊢ ((1o ≼ 2o ∧ 𝐴 ∈ V) → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
7	3, 5, 6	sylancr 598	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
8		canthp1lem1 10632	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 2o) ≼ 𝒫 𝐴)
9		domtr 9000	. . 3 ⊢ (((𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o) ∧ (𝐴 ⊔ 2o) ≼ 𝒫 𝐴) → (𝐴 ⊔ 1o) ≼ 𝒫 𝐴)
10	7, 8, 9	syl2anc 595	. 2 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≼ 𝒫 𝐴)
11		fal 1584	. . 3 ⊢ ¬ ⊥
12		ensym 8996	. . . . 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 221	. . . 4 ⊢ ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
15		f1of 6820	. . . . . . . . . 10 ⊢ (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → 𝑓:𝒫 𝐴⟶(𝐴 ⊔ 1o))
16		pwidg 4582	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
17	5, 16	syl 18	. . . . . . . . . 10 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ 𝒫 𝐴)
18		ffvelcdm 7076	. . . . . . . . . 10 ⊢ ((𝑓:𝒫 𝐴⟶(𝐴 ⊔ 1o) ∧ 𝐴 ∈ 𝒫 𝐴) → (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o))
19	15, 17, 18	syl2anr 608	. . . . . . . . 9 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o))
20		dju1dif 10152	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
21	5, 19, 20	syl2an2r 697	. . . . . . . 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 221	. . . . . . 7 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ∃𝑔 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
24		simpll 778	. . . . . . . . 9 ⊢ (((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 1o ≺ 𝐴)
25		simplr 780	. . . . . . . . 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 489	. . . . . . . . 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 2767	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝐴 ↔ 𝑥 = 𝐴))
28		id 23	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥)
29	27, 28	ifbieq2d 4514	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → if(𝑤 = 𝐴, ∅, 𝑤) = if(𝑥 = 𝐴, ∅, 𝑥))
30	29	cbvmptv 5215	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)) = (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))
31	30	coeq2i 5846	. . . . . . . . 9 ⊢ ((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤))) = ((𝑔 ∘ 𝑓) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
32		eqid 2763	. . . . . . . . . 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 10624	. . . . . . . . 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 2763	. . . . . . . . 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 10633	. . . . . . . 8 ⊢ ¬ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
36	35	pm2.21i 120	. . . . . . 7 ⊢ (((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → ⊥)
37	23, 36	exlimddv 1965	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ⊥)
38	37	ex 417	. . . . 5 ⊢ (1o ≺ 𝐴 → (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → ⊥))
39	38	exlimdv 1963	. . . 4 ⊢ (1o ≺ 𝐴 → (∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → ⊥))
40	14, 39	syl5 35	. . 3 ⊢ (1o ≺ 𝐴 → ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → ⊥))
41	11, 40	mtoi 202	. 2 ⊢ (1o ≺ 𝐴 → ¬ (𝐴 ⊔ 1o) ≈ 𝒫 𝐴)
42		brsdom 8967	. 2 ⊢ ((𝐴 ⊔ 1o) ≺ 𝒫 𝐴 ↔ ((𝐴 ⊔ 1o) ≼ 𝒫 𝐴 ∧ ¬ (𝐴 ⊔ 1o) ≈ 𝒫 𝐴))
43	10, 41, 42	sylanbrc 594	1 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≺ 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570  ⊥wfal 1582  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  ifcif 4487  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872   class class class wbr 5109  {copab 5173   ↦ cmpt 5192   We wwe 5613   × cxp 5659  ^◡ccnv 5660  dom cdm 5661   “ cima 5664   ∘ ccom 5665  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  1_oc1o 8442  2_oc2o 8443   ≈ cen 8936   ≼ cdom 8937   ≺ csdm 8938   ⊔ cdju 9880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9468  df-dju 9883  df-card 9921

This theorem is referenced by:  finngch  10635  gchdju1  10636