Theorem canthnum 10689

Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9170. 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 5378	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		inex1g 5319	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3		infpwfidom 10068	. . . 4 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5		inex1g 5319	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
6	1, 5	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7		finnum 9988	. . . . . 6 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
8	7	ssriv 3987	. . . . 5 ⊢ Fin ⊆ dom card
9		sslin 4243	. . . . 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 9040	. . . 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 9047	. . 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 2737	. . . . . . 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 10684	. . . . . 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 2737	. . . . . 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 2737	. . . . . 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 10688	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
20		f1of1 6847	. . . . 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 1936	. . 3 ⊢ (𝐴 ∈ 𝑉 → ¬ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
23		ensym 9043	. . . 4 ⊢ (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24		bren 8995	. . . 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 9015	. 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 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  {csn 4626  ∪ cuni 4907   class class class wbr 5143  {copab 5205   We wwe 5636   × cxp 5683  ^◡ccnv 5684  dom cdm 5685   “ cima 5688  –1-1→wf1 6558  –1-1-onto→wf1o 6560  ‘cfv 6561   ≈ cen 8982   ≼ cdom 8983   ≺ csdm 8984  Fincfn 8985  cardccrd 9975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-card 9979

This theorem is referenced by: (None)