Theorem canthnum 10537

Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9043. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)

Assertion

Ref	Expression
canthnum	⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ (𝒫 𝐴 ∩ dom card))

Proof of Theorem canthnum

Dummy variables 𝑓 𝑎 𝑟 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 5316	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		inex1g 5257	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3		infpwfidom 9916	. . . 4 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5		inex1g 5257	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
6	1, 5	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7		finnum 9838	. . . . . 6 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
8	7	ssriv 3938	. . . . 5 ⊢ Fin ⊆ dom card
9		sslin 4193	. . . . 5 ⊢ (Fin ⊆ dom card → (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐴 ∩ dom card))
10	8, 9	ax-mp 5	. . . 4 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐴 ∩ dom card)
11		ssdomg 8922	. . . 4 ⊢ ((𝒫 𝐴 ∩ dom card) ∈ V → ((𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)))
12	6, 10, 11	mpisyl 21	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card))
13		domtr 8929	. . 3 ⊢ ((𝐴 ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)) → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
14	4, 12, 13	syl2anc 584	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
15		eqid 2731	. . . . . . 7 ⊢ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))}
16	15	fpwwecbv 10532	. . . . . 6 ⊢ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (𝑓‘(◡𝑠 “ {𝑧})) = 𝑧))}
17		eqid 2731	. . . . . 6 ⊢ ∪ dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))} = ∪ dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))}
18		eqid 2731	. . . . . 6 ⊢ (◡({⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))}‘∪ dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))}) “ {(𝑓‘∪ dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))})}) = (◡({⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))}‘∪ dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))}) “ {(𝑓‘∪ dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘(◡𝑟 “ {𝑦})) = 𝑦))})})
19	16, 17, 18	canthnumlem 10536	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
20		f1of1 6762	. . . . 5 ⊢ (𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴 → 𝑓:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
21	19, 20	nsyl 140	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
22	21	nexdv 1937	. . 3 ⊢ (𝐴 ∈ 𝑉 → ¬ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
23		ensym 8925	. . . 4 ⊢ (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24		bren 8879	. . . 4 ⊢ ((𝒫 𝐴 ∩ dom card) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
25	23, 24	sylib 218	. . 3 ⊢ (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
26	22, 25	nsyl 140	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card))
27		brsdom 8897	. 2 ⊢ (𝐴 ≺ (𝒫 𝐴 ∩ dom card) ↔ (𝐴 ≼ (𝒫 𝐴 ∩ dom card) ∧ ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card)))
28	14, 26, 27	sylanbrc 583	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ (𝒫 𝐴 ∩ dom card))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4550  {csn 4576  ∪ cuni 4859   class class class wbr 5091  {copab 5153   We wwe 5568   × cxp 5614  ^◡ccnv 5615  dom cdm 5616   “ cima 5619  –1-1→wf1 6478  –1-1-onto→wf1o 6480  ‘cfv 6481   ≈ cen 8866   ≼ cdom 8867   ≺ csdm 8868  Fincfn 8869  cardccrd 9825

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-oi 9396  df-card 9829

This theorem is referenced by: (None)