Theorem scancan 6332

Description: Strongly Cantorian implies Cantorian. Observation from [Holmes], p. 134. (Contributed by Scott Fenton, 19-Apr-2021.)

Assertion

Ref	Expression
scancan	⊢ (A ∈ SCan → A ∈ Can )

Proof of Theorem scancan

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4112	. . . . . 6 ⊢ {x} ∈ V
2		eqid 2353	. . . . . 6 ⊢ (x ∈ A ↦ {x}) = (x ∈ A ↦ {x})
3	1, 2	fnmpti 5691	. . . . 5 ⊢ (x ∈ A ↦ {x}) Fn A
4		elpw1 4145	. . . . . . 7 ⊢ (y ∈ ℘1A ↔ ∃z ∈ A y = {z})
5		euequ1 2292	. . . . . . . . 9 ⊢ ∃!x x = z
6		eqeq1 2359	. . . . . . . . . . 11 ⊢ (y = {z} → (y = {x} ↔ {z} = {x}))
7		vex 2863	. . . . . . . . . . . . 13 ⊢ z ∈ V
8	7	sneqb 3877	. . . . . . . . . . . 12 ⊢ ({z} = {x} ↔ z = x)
9		equcom 1680	. . . . . . . . . . . 12 ⊢ (z = x ↔ x = z)
10	8, 9	bitri 240	. . . . . . . . . . 11 ⊢ ({z} = {x} ↔ x = z)
11	6, 10	syl6bb 252	. . . . . . . . . 10 ⊢ (y = {z} → (y = {x} ↔ x = z))
12	11	eubidv 2212	. . . . . . . . 9 ⊢ (y = {z} → (∃!x y = {x} ↔ ∃!x x = z))
13	5, 12	mpbiri 224	. . . . . . . 8 ⊢ (y = {z} → ∃!x y = {x})
14	13	rexlimivw 2735	. . . . . . 7 ⊢ (∃z ∈ A y = {z} → ∃!x y = {x})
15	4, 14	sylbi 187	. . . . . 6 ⊢ (y ∈ ℘1A → ∃!x y = {x})
16		df-mpt 5653	. . . . . . . 8 ⊢ (x ∈ A ↦ {x}) = {⟨x, y⟩ ∣ (x ∈ A ∧ y = {x})}
17	16	cnveqi 4888	. . . . . . 7 ⊢ ◡(x ∈ A ↦ {x}) = ◡{⟨x, y⟩ ∣ (x ∈ A ∧ y = {x})}
18		cnvopab 5031	. . . . . . 7 ⊢ ◡{⟨x, y⟩ ∣ (x ∈ A ∧ y = {x})} = {⟨y, x⟩ ∣ (x ∈ A ∧ y = {x})}
19		eleq1 2413	. . . . . . . . . 10 ⊢ (y = {x} → (y ∈ ℘1A ↔ {x} ∈ ℘1A))
20		snelpw1 4147	. . . . . . . . . 10 ⊢ ({x} ∈ ℘1A ↔ x ∈ A)
21	19, 20	syl6rbb 253	. . . . . . . . 9 ⊢ (y = {x} → (x ∈ A ↔ y ∈ ℘1A))
22	21	pm5.32ri 619	. . . . . . . 8 ⊢ ((x ∈ A ∧ y = {x}) ↔ (y ∈ ℘1A ∧ y = {x}))
23	22	opabbii 4627	. . . . . . 7 ⊢ {⟨y, x⟩ ∣ (x ∈ A ∧ y = {x})} = {⟨y, x⟩ ∣ (y ∈ ℘1A ∧ y = {x})}
24	17, 18, 23	3eqtri 2377	. . . . . 6 ⊢ ◡(x ∈ A ↦ {x}) = {⟨y, x⟩ ∣ (y ∈ ℘1A ∧ y = {x})}
25	15, 24	fnopab 5208	. . . . 5 ⊢ ◡(x ∈ A ↦ {x}) Fn ℘1A
26		dff1o4 5295	. . . . 5 ⊢ ((x ∈ A ↦ {x}):A–1-1-onto→℘1A ↔ ((x ∈ A ↦ {x}) Fn A ∧ ◡(x ∈ A ↦ {x}) Fn ℘1A))
27	3, 25, 26	mpbir2an 886	. . . 4 ⊢ (x ∈ A ↦ {x}):A–1-1-onto→℘1A
28		f1oeng 6033	. . . 4 ⊢ (((x ∈ A ↦ {x}) ∈ V ∧ (x ∈ A ↦ {x}):A–1-1-onto→℘1A) → A ≈ ℘1A)
29	27, 28	mpan2 652	. . 3 ⊢ ((x ∈ A ↦ {x}) ∈ V → A ≈ ℘1A)
30		ensymi 6037	. . 3 ⊢ (A ≈ ℘1A → ℘1A ≈ A)
31	29, 30	syl 15	. 2 ⊢ ((x ∈ A ↦ {x}) ∈ V → ℘1A ≈ A)
32		elscan 6331	. 2 ⊢ (A ∈ SCan ↔ (x ∈ A ↦ {x}) ∈ V)
33		elcan 6330	. 2 ⊢ (A ∈ Can ↔ ℘1A ≈ A)
34	31, 32, 33	3imtr4i 257	1 ⊢ (A ∈ SCan → A ∈ Can )

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃wrex 2616  Vcvv 2860  {csn 3738  ℘₁cpw1 4136  {copab 4623   class class class wbr 4640  ^◡ccnv 4772   Fn wfn 4777  –1-1-onto→wf1o 4781   ↦ cmpt 5652   ≈ cen 6029   Can ccan 6324   SCan cscan 6326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-swap 4725  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-mpt 5653  df-en 6030  df-can 6325  df-scan 6327

This theorem is referenced by: (None)