Theorem scancan 6331

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

Assertion

Ref	Expression
scancan	|- (A e. SCan -> A e. 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 4111	. . . . . 6 |- {x} e. _V
2		eqid 2353	. . . . . 6 |- (x e. A |-> {x}) = (x e. A |-> {x})
3	1, 2	fnmpti 5690	. . . . 5 |- (x e. A |-> {x}) Fn A
4		elpw1 4144	. . . . . . 7 |- (y e. ~P1 A <-> E.z e. A y = {z})
5		euequ1 2292	. . . . . . . . 9 |- E!x x = z
6		eqeq1 2359	. . . . . . . . . . 11 |- (y = {z} -> (y = {x} <-> {z} = {x}))
7		vex 2862	. . . . . . . . . . . . 13 |- z e. _V
8	7	sneqb 3876	. . . . . . . . . . . 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} -> (E!x y = {x} <-> E!x x = z))
13	5, 12	mpbiri 224	. . . . . . . 8 |- (y = {z} -> E!x y = {x})
14	13	rexlimivw 2734	. . . . . . 7 |- (E.z e. A y = {z} -> E!x y = {x})
15	4, 14	sylbi 187	. . . . . 6 |- (y e. ~P1 A -> E!x y = {x})
16		df-mpt 5652	. . . . . . . 8 |- (x e. A |-> {x}) = {<.x, y>. | (x e. A /\ y = {x})}
17	16	cnveqi 4887	. . . . . . 7 |- `'(x e. A |-> {x}) = `'{<.x, y>. | (x e. A /\ y = {x})}
18		cnvopab 5030	. . . . . . 7 |- `'{<.x, y>. | (x e. A /\ y = {x})} = {<.y, x>. | (x e. A /\ y = {x})}
19		eleq1 2413	. . . . . . . . . 10 |- (y = {x} -> (y e. ~P1 A <-> {x} e. ~P1 A))
20		snelpw1 4146	. . . . . . . . . 10 |- ({x} e. ~P1 A <-> x e. A)
21	19, 20	syl6rbb 253	. . . . . . . . 9 |- (y = {x} -> (x e. A <-> y e. ~P1 A))
22	21	pm5.32ri 619	. . . . . . . 8 |- ((x e. A /\ y = {x}) <-> (y e. ~P1 A /\ y = {x}))
23	22	opabbii 4626	. . . . . . 7 |- {<.y, x>. | (x e. A /\ y = {x})} = {<.y, x>. | (y e. ~P1 A /\ y = {x})}
24	17, 18, 23	3eqtri 2377	. . . . . 6 |- `'(x e. A |-> {x}) = {<.y, x>. | (y e. ~P1 A /\ y = {x})}
25	15, 24	fnopab 5207	. . . . 5 |- `'(x e. A |-> {x}) Fn ~P1 A
26		dff1o4 5294	. . . . 5 |- ((x e. A |-> {x}):A-1-1-onto->~P1 A <-> ((x e. A |-> {x}) Fn A /\ `'(x e. A |-> {x}) Fn ~P1 A))
27	3, 25, 26	mpbir2an 886	. . . 4 |- (x e. A |-> {x}):A-1-1-onto->~P1 A
28		f1oeng 6032	. . . 4 |- (((x e. A |-> {x}) e. _V /\ (x e. A |-> {x}):A-1-1-onto->~P1 A) -> A ~~ ~P1 A)
29	27, 28	mpan2 652	. . 3 |- ((x e. A |-> {x}) e. _V -> A ~~ ~P1 A)
30		ensymi 6036	. . 3 |- (A ~~ ~P1 A -> ~P1 A ~~ A)
31	29, 30	syl 15	. 2 |- ((x e. A |-> {x}) e. _V -> ~P1 A ~~ A)
32		elscan 6330	. 2 |- (A e. SCan <-> (x e. A |-> {x}) e. _V)
33		elcan 6329	. 2 |- (A e. Can <-> ~P1 A ~~ A)
34	31, 32, 33	3imtr4i 257	1 |- (A e. SCan -> A e. Can )

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  E!weu 2204  E.wrex 2615  _Vcvv 2859  {csn 3737  ~P₁ cpw1 4135  {copab 4622   class class class wbr 4639  `'ccnv 4771   Fn wfn 4776  -1-1-onto->wf1o 4780   |-> cmpt 5651   ~~ cen 6028   Can ccan 6323   SCan cscan 6325

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-swap 4724  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-mpt 5652  df-en 6029  df-can 6324  df-scan 6326

This theorem is referenced by: (None)