Theorem canthp1 9911

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 8553	. . . . 5 ⊢ 1o ≺ 2o
2		sdomdom 8375	. . . . 5 ⊢ (1o ≺ 2o → 1o ≼ 2o)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1o ≼ 2o
4		relsdom 8354	. . . . 5 ⊢ Rel ≺
5	4	brrelex2i 5487	. . . 4 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
6		djudom2 9444	. . . 4 ⊢ ((1o ≼ 2o ∧ 𝐴 ∈ V) → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
7	3, 5, 6	sylancr 587	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o))
8		canthp1lem1 9909	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 2o) ≼ 𝒫 𝐴)
9		domtr 8400	. . 3 ⊢ (((𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 2o) ∧ (𝐴 ⊔ 2o) ≼ 𝒫 𝐴) → (𝐴 ⊔ 1o) ≼ 𝒫 𝐴)
10	7, 8, 9	syl2anc 584	. 2 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≼ 𝒫 𝐴)
11		fal 1534	. . 3 ⊢ ¬ ⊥
12		ensym 8396	. . . . 5 ⊢ ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → 𝒫 𝐴 ≈ (𝐴 ⊔ 1o))
13		bren 8356	. . . . 5 ⊢ (𝒫 𝐴 ≈ (𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
14	12, 13	sylib 219	. . . 4 ⊢ ((𝐴 ⊔ 1o) ≈ 𝒫 𝐴 → ∃𝑓 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))
15		f1of 6475	. . . . . . . . . 10 ⊢ (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → 𝑓:𝒫 𝐴⟶(𝐴 ⊔ 1o))
16		pwidg 4462	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
17	5, 16	syl 17	. . . . . . . . . 10 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ 𝒫 𝐴)
18		ffvelrn 6705	. . . . . . . . . 10 ⊢ ((𝑓:𝒫 𝐴⟶(𝐴 ⊔ 1o) ∧ 𝐴 ∈ 𝒫 𝐴) → (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o))
19	15, 17, 18	syl2anr 596	. . . . . . . . 9 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o))
20		dju1dif 9433	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑓‘𝐴) ∈ (𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
21	5, 19, 20	syl2an2r 681	. . . . . . . 8 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴)
22		bren 8356	. . . . . . . 8 ⊢ (((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)}) ≈ 𝐴 ↔ ∃𝑔 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
23	21, 22	sylib 219	. . . . . . 7 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ∃𝑔 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴)
24		simpll 763	. . . . . . . . 9 ⊢ (((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) ∧ 𝑔:((𝐴 ⊔ 1o) ∖ {(𝑓‘𝐴)})–1-1-onto→𝐴) → 1o ≺ 𝐴)
25		simplr 765	. . . . . . . . 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 485	. . . . . . . . 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 2797	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝐴 ↔ 𝑥 = 𝐴))
28		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥)
29	27, 28	ifbieq2d 4400	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → if(𝑤 = 𝐴, ∅, 𝑤) = if(𝑥 = 𝐴, ∅, 𝑥))
30	29	cbvmptv 5055	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤)) = (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))
31	30	coeq2i 5609	. . . . . . . . 9 ⊢ ((𝑔 ∘ 𝑓) ∘ (𝑤 ∈ 𝒫 𝐴 ↦ if(𝑤 = 𝐴, ∅, 𝑤))) = ((𝑔 ∘ 𝑓) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
32		eqid 2793	. . . . . . . . . 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 9901	. . . . . . . . 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 2793	. . . . . . . . 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 9910	. . . . . . . 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 1911	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) → ⊥)
38	37	ex 413	. . . . 5 ⊢ (1o ≺ 𝐴 → (𝑓:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o) → ⊥))
39	38	exlimdv 1909	. . . 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 200	. 2 ⊢ (1o ≺ 𝐴 → ¬ (𝐴 ⊔ 1o) ≈ 𝒫 𝐴)
42		brsdom 8370	. 2 ⊢ ((𝐴 ⊔ 1o) ≺ 𝒫 𝐴 ↔ ((𝐴 ⊔ 1o) ≼ 𝒫 𝐴 ∧ ¬ (𝐴 ⊔ 1o) ≈ 𝒫 𝐴))
43	10, 41, 42	sylanbrc 583	1 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≺ 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1520  ⊥wfal 1532  ∃wex 1759   ∈ wcel 2079  ∀wral 3103  Vcvv 3432   ∖ cdif 3851   ⊆ wss 3854  ∅c0 4206  ifcif 4375  𝒫 cpw 4447  {csn 4466  ∪ cuni 4739   class class class wbr 4956  {copab 5018   ↦ cmpt 5035   We wwe 5393   × cxp 5433  ^◡ccnv 5434  dom cdm 5435   “ cima 5438   ∘ ccom 5439  ⟶wf 6213  –1-1-onto→wf1o 6216  ‘cfv 6217  1_oc1o 7937  2_oc2o 7938   ≈ cen 8344   ≼ cdom 8345   ≺ csdm 8346   ⊔ cdju 9162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-inf2 8939

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-fal 1533  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-se 5395  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-isom 6226  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-2o 7945  df-oadd 7948  df-er 8130  df-map 8249  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-oi 8810  df-dju 9165  df-card 9203

This theorem is referenced by:  finngch  9912  gchdju1  9913