Theorem canthnum 10604

Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9098. 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 5334	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		inex1g 5274	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3		infpwfidom 9981	. . . 4 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5		inex1g 5274	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
6	1, 5	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7		finnum 9903	. . . . . 6 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
8	7	ssriv 3940	. . . . 5 ⊢ Fin ⊆ dom card
9		sslin 4194	. . . . 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 8977	. . . 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 8984	. . 3 ⊢ ((𝐴 ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)) → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
14	4, 12, 13	syl2anc 593	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
15		eqid 2761	. . . . . . 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 10599	. . . . . 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 2761	. . . . . 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 2761	. . . . . 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 10603	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
20		f1of1 6801	. . . . 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 1955	. . 3 ⊢ (𝐴 ∈ 𝑉 → ¬ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
23		ensym 8980	. . . 4 ⊢ (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24		bren 8933	. . . 4 ⊢ ((𝒫 𝐴 ∩ dom card) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto→𝐴)
25	23, 24	sylib 220	. . 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 8951	. 2 ⊢ (𝐴 ≺ (𝒫 𝐴 ∩ dom card) ↔ (𝐴 ≼ (𝒫 𝐴 ∩ dom card) ∧ ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card)))
28	14, 26, 27	sylanbrc 592	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ (𝒫 𝐴 ∩ dom card))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4554  {csn 4581  ∪ cuni 4864   class class class wbr 5099  {copab 5161   We wwe 5597   × cxp 5643  ^◡ccnv 5644  dom cdm 5645   “ cima 5648  –1-1→wf1 6514  –1-1-onto→wf1o 6516  ‘cfv 6517   ≈ cen 8920   ≼ cdom 8921   ≺ csdm 8922  Fincfn 8923  cardccrd 9890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-oi 9455  df-card 9894

This theorem is referenced by: (None)