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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 SCanA Can )

Proof of Theorem scancan
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4111 . . . . . 6 {x} V
2 eqid 2353 . . . . . 6 (x A {x}) = (x A {x})
31, 2fnmpti 5690 . . . . 5 (x A {x}) Fn A
4 elpw1 4144 . . . . . . 7 (y 1Az A y = {z})
5 euequ1 2292 . . . . . . . . 9 ∃!x x = z
6 eqeq1 2359 . . . . . . . . . . 11 (y = {z} → (y = {x} ↔ {z} = {x}))
7 vex 2862 . . . . . . . . . . . . 13 z V
87sneqb 3876 . . . . . . . . . . . 12 ({z} = {x} ↔ z = x)
9 equcom 1680 . . . . . . . . . . . 12 (z = xx = z)
108, 9bitri 240 . . . . . . . . . . 11 ({z} = {x} ↔ x = z)
116, 10syl6bb 252 . . . . . . . . . 10 (y = {z} → (y = {x} ↔ x = z))
1211eubidv 2212 . . . . . . . . 9 (y = {z} → (∃!x y = {x} ↔ ∃!x x = z))
135, 12mpbiri 224 . . . . . . . 8 (y = {z} → ∃!x y = {x})
1413rexlimivw 2734 . . . . . . 7 (z A y = {z} → ∃!x y = {x})
154, 14sylbi 187 . . . . . 6 (y 1A∃!x y = {x})
16 df-mpt 5652 . . . . . . . 8 (x A {x}) = {x, y (x A y = {x})}
1716cnveqi 4887 . . . . . . 7 (x A {x}) = {x, y (x A y = {x})}
18 cnvopab 5030 . . . . . . 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 4146 . . . . . . . . . 10 ({x} 1Ax A)
2119, 20syl6rbb 253 . . . . . . . . 9 (y = {x} → (x Ay 1A))
2221pm5.32ri 619 . . . . . . . 8 ((x A y = {x}) ↔ (y 1A y = {x}))
2322opabbii 4626 . . . . . . 7 {y, x (x A y = {x})} = {y, x (y 1A y = {x})}
2417, 18, 233eqtri 2377 . . . . . 6 (x A {x}) = {y, x (y 1A y = {x})}
2515, 24fnopab 5207 . . . . 5 (x A {x}) Fn 1A
26 dff1o4 5294 . . . . 5 ((x A {x}):A1-1-onto1A ↔ ((x A {x}) Fn A (x A {x}) Fn 1A))
273, 25, 26mpbir2an 886 . . . 4 (x A {x}):A1-1-onto1A
28 f1oeng 6032 . . . 4 (((x A {x}) V (x A {x}):A1-1-onto1A) → A1A)
2927, 28mpan2 652 . . 3 ((x A {x}) V → A1A)
30 ensymi 6036 . . 3 (A1A1AA)
3129, 30syl 15 . 2 ((x A {x}) V → 1AA)
32 elscan 6330 . 2 (A SCan ↔ (x A {x}) V)
33 elcan 6329 . 2 (A Can1AA)
3431, 32, 333imtr4i 257 1 (A SCanA Can )
Colors of variables: wff setvar class
Syntax hints:  wi 4   wa 358   = wceq 1642   wcel 1710  ∃!weu 2204  wrex 2615  Vcvv 2859  {csn 3737  1cpw1 4135  {copab 4622   class class class wbr 4639  ccnv 4771   Fn wfn 4776  1-1-ontowf1o 4780   cmpt 5651  cen 6028   Can ccan 6323   SCan cscan 6325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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)
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