Theorem canthnum 10562

Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9054. 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 5320	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		inex1g 5261	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3		infpwfidom 9941	. . . 4 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5		inex1g 5261	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
6	1, 5	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7		finnum 9863	. . . . . 6 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
8	7	ssriv 3941	. . . . 5 ⊢ Fin ⊆ dom card
9		sslin 4196	. . . . 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 8932	. . . 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 8939	. . 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 2729	. . . . . . 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 10557	. . . . . 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 2729	. . . . . 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 2729	. . . . . 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 10561	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
20		f1of1 6767	. . . . 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 8935	. . . 4 ⊢ (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24		bren 8889	. . . 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 8907	. 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 2109  ∀wral 3044  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4553  {csn 4579  ∪ cuni 4861   class class class wbr 5095  {copab 5157   We wwe 5575   × cxp 5621  ^◡ccnv 5622  dom cdm 5623   “ cima 5626  –1-1→wf1 6483  –1-1-onto→wf1o 6485  ‘cfv 6486   ≈ cen 8876   ≼ cdom 8877   ≺ csdm 8878  Fincfn 8879  cardccrd 9850

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  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 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-oi 9421  df-card 9854

This theorem is referenced by: (None)