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Theorem csucex 6259
 Description: The function mapping x to its cardinal successor exists. (Contributed by Scott Fenton, 30-Jul-2019.)
Assertion
Ref Expression
csucex (x V (x +c 1c)) V

Proof of Theorem csucex
Dummy variables y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brcnv 4892 . . . . . . . . . 10 (y1st ww1st y)
2 vex 2862 . . . . . . . . . . 11 y V
32br1st 4858 . . . . . . . . . 10 (w1st yx w = y, x)
41, 3bitri 240 . . . . . . . . 9 (y1st wx w = y, x)
54anbi1i 676 . . . . . . . 8 ((y1st w w( AddC (V × {1c}))z) ↔ (x w = y, x w( AddC (V × {1c}))z))
6 19.41v 1901 . . . . . . . 8 (x(w = y, x w( AddC (V × {1c}))z) ↔ (x w = y, x w( AddC (V × {1c}))z))
75, 6bitr4i 243 . . . . . . 7 ((y1st w w( AddC (V × {1c}))z) ↔ x(w = y, x w( AddC (V × {1c}))z))
87exbii 1582 . . . . . 6 (w(y1st w w( AddC (V × {1c}))z) ↔ wx(w = y, x w( AddC (V × {1c}))z))
9 excom 1741 . . . . . . 7 (wx(w = y, x w( AddC (V × {1c}))z) ↔ xw(w = y, x w( AddC (V × {1c}))z))
10 vex 2862 . . . . . . . . . 10 x V
112, 10opex 4588 . . . . . . . . 9 y, x V
12 breq1 4642 . . . . . . . . . 10 (w = y, x → (w( AddC (V × {1c}))zy, x( AddC (V × {1c}))z))
13 brres 4949 . . . . . . . . . . 11 (y, x( AddC (V × {1c}))z ↔ (y, x AddC z y, x (V × {1c})))
142, 10braddcfn 5826 . . . . . . . . . . . 12 (y, x AddC z ↔ (y +c x) = z)
15 opelxp 4811 . . . . . . . . . . . . . 14 (y, x (V × {1c}) ↔ (y V x {1c}))
162, 15mpbiran 884 . . . . . . . . . . . . 13 (y, x (V × {1c}) ↔ x {1c})
17 elsn 3748 . . . . . . . . . . . . 13 (x {1c} ↔ x = 1c)
1816, 17bitri 240 . . . . . . . . . . . 12 (y, x (V × {1c}) ↔ x = 1c)
1914, 18anbi12ci 679 . . . . . . . . . . 11 ((y, x AddC z y, x (V × {1c})) ↔ (x = 1c (y +c x) = z))
2013, 19bitri 240 . . . . . . . . . 10 (y, x( AddC (V × {1c}))z ↔ (x = 1c (y +c x) = z))
2112, 20syl6bb 252 . . . . . . . . 9 (w = y, x → (w( AddC (V × {1c}))z ↔ (x = 1c (y +c x) = z)))
2211, 21ceqsexv 2894 . . . . . . . 8 (w(w = y, x w( AddC (V × {1c}))z) ↔ (x = 1c (y +c x) = z))
2322exbii 1582 . . . . . . 7 (xw(w = y, x w( AddC (V × {1c}))z) ↔ x(x = 1c (y +c x) = z))
249, 23bitri 240 . . . . . 6 (wx(w = y, x w( AddC (V × {1c}))z) ↔ x(x = 1c (y +c x) = z))
258, 24bitri 240 . . . . 5 (w(y1st w w( AddC (V × {1c}))z) ↔ x(x = 1c (y +c x) = z))
26 1cex 4142 . . . . . 6 1c V
27 addceq2 4384 . . . . . . 7 (x = 1c → (y +c x) = (y +c 1c))
2827eqeq1d 2361 . . . . . 6 (x = 1c → ((y +c x) = z ↔ (y +c 1c) = z))
2926, 28ceqsexv 2894 . . . . 5 (x(x = 1c (y +c x) = z) ↔ (y +c 1c) = z)
3025, 29bitri 240 . . . 4 (w(y1st w w( AddC (V × {1c}))z) ↔ (y +c 1c) = z)
31 opelco 4884 . . . 4 (y, z (( AddC (V × {1c})) 1st ) ↔ w(y1st w w( AddC (V × {1c}))z))
32 mptv 5718 . . . . . 6 (x V (x +c 1c)) = {x, w w = (x +c 1c)}
3332eleq2i 2417 . . . . 5 (y, z (x V (x +c 1c)) ↔ y, z {x, w w = (x +c 1c)})
34 vex 2862 . . . . . 6 z V
35 addceq1 4383 . . . . . . 7 (x = y → (x +c 1c) = (y +c 1c))
3635eqeq2d 2364 . . . . . 6 (x = y → (w = (x +c 1c) ↔ w = (y +c 1c)))
37 eqeq1 2359 . . . . . . 7 (w = z → (w = (y +c 1c) ↔ z = (y +c 1c)))
38 eqcom 2355 . . . . . . 7 (z = (y +c 1c) ↔ (y +c 1c) = z)
3937, 38syl6bb 252 . . . . . 6 (w = z → (w = (y +c 1c) ↔ (y +c 1c) = z))
402, 34, 36, 39opelopab 4708 . . . . 5 (y, z {x, w w = (x +c 1c)} ↔ (y +c 1c) = z)
4133, 40bitri 240 . . . 4 (y, z (x V (x +c 1c)) ↔ (y +c 1c) = z)
4230, 31, 413bitr4ri 269 . . 3 (y, z (x V (x +c 1c)) ↔ y, z (( AddC (V × {1c})) 1st ))
4342eqrelriv 4850 . 2 (x V (x +c 1c)) = (( AddC (V × {1c})) 1st )