Theorem canthnum 9471

Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8113. 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 4850	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		inex1g 4801	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3		infpwfidom 8851	. . . 4 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5		inex1g 4801	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
6	1, 5	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7		finnum 8774	. . . . . 6 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
8	7	ssriv 3607	. . . . 5 ⊢ Fin ⊆ dom card
9		sslin 3839	. . . . 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 8001	. . . 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 8009	. . 3 ⊢ ((𝐴 ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)) → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
14	4, 12, 13	syl2anc 693	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
15		eqid 2622	. . . . . . 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 9466	. . . . . 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 2622	. . . . . 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 2622	. . . . . 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 9470	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
20		f1of1 6136	. . . . 5 ⊢ (𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴 → 𝑓:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
21	19, 20	nsyl 135	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
22	21	nexdv 1864	. . 3 ⊢ (𝐴 ∈ 𝑉 → ¬ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
23		ensym 8005	. . . 4 ⊢ (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24		bren 7964	. . . 4 ⊢ ((𝒫 𝐴 ∩ dom card) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
25	23, 24	sylib 208	. . 3 ⊢ (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
26	22, 25	nsyl 135	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card))
27		brsdom 7978	. 2 ⊢ (𝐴 ≺ (𝒫 𝐴 ∩ dom card) ↔ (𝐴 ≼ (𝒫 𝐴 ∩ dom card) ∧ ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card)))
28	14, 26, 27	sylanbrc 698	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ (𝒫 𝐴 ∩ dom card))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   class class class wbr 4653  {copab 4712   We wwe 5072   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-om 7066  df-1st 7168  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765

This theorem is referenced by: (None)