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Theorem lecex 6115
 Description: Cardinal less than or equal is a set. (Contributed by SF, 24-Feb-2015.)
Assertion
Ref Expression
lecex c V

Proof of Theorem lecex
Dummy variables a b t u x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r2ex 2652 . . . . 5 (x a y b x yxy((x a y b) x y))
2 19.41vv 1902 . . . . . . 7 (tu(((t S a u S b) (t = {x} u = {y})) x y) ↔ (tu((t S a u S b) (t = {x} u = {y})) x y))
3 anass 630 . . . . . . . 8 ((((t S a u S b) (t = {x} u = {y})) x y) ↔ ((t S a u S b) ((t = {x} u = {y}) x y)))
432exbii 1583 . . . . . . 7 (tu(((t S a u S b) (t = {x} u = {y})) x y) ↔ tu((t S a u S b) ((t = {x} u = {y}) x y)))
5 ancom 437 . . . . . . . . . . 11 (((t S a u S b) (t = {x} u = {y})) ↔ ((t = {x} u = {y}) (t S a u S b)))
6 df-3an 936 . . . . . . . . . . 11 ((t = {x} u = {y} (t S a u S b)) ↔ ((t = {x} u = {y}) (t S a u S b)))
75, 6bitr4i 243 . . . . . . . . . 10 (((t S a u S b) (t = {x} u = {y})) ↔ (t = {x} u = {y} (t S a u S b)))
872exbii 1583 . . . . . . . . 9 (tu((t S a u S b) (t = {x} u = {y})) ↔ tu(t = {x} u = {y} (t S a u S b)))
9 snex 4111 . . . . . . . . . 10 {x} V
10 snex 4111 . . . . . . . . . 10 {y} V
11 breq1 4642 . . . . . . . . . . 11 (t = {x} → (t S a ↔ {x} S a))
1211anbi1d 685 . . . . . . . . . 10 (t = {x} → ((t S a u S b) ↔ ({x} S a u S b)))
13 breq1 4642 . . . . . . . . . . 11 (u = {y} → (u S b ↔ {y} S b))
1413anbi2d 684 . . . . . . . . . 10 (u = {y} → (({x} S a u S b) ↔ ({x} S a {y} S b)))
159, 10, 12, 14ceqsex2v 2896 . . . . . . . . 9 (tu(t = {x} u = {y} (t S a u S b)) ↔ ({x} S a {y} S b))
16 vex 2862 . . . . . . . . . . 11 x V
17 vex 2862 . . . . . . . . . . 11 a V
1816, 17brssetsn 4759 . . . . . . . . . 10 ({x} S ax a)
19 vex 2862 . . . . . . . . . . 11 y V
20 vex 2862 . . . . . . . . . . 11 b V
2119, 20brssetsn 4759 . . . . . . . . . 10 ({y} S by b)
2218, 21anbi12i 678 . . . . . . . . 9 (({x} S a {y} S b) ↔ (x a y b))
238, 15, 223bitri 262 . . . . . . . 8 (tu((t S a u S b) (t = {x} u = {y})) ↔ (x a y b))
2423anbi1i 676 . . . . . . 7 ((tu((t S a u S b) (t = {x} u = {y})) x y) ↔ ((x a y b) x y))
252, 4, 243bitr3i 266 . . . . . 6 (tu((t S a u S b) ((t = {x} u = {y}) x y)) ↔ ((x a y b) x y))
26252exbii 1583 . . . . 5 (xytu((t S a u S b) ((t = {x} u = {y}) x y)) ↔ xy((x a y b) x y))
271, 26bitr4i 243 . . . 4 (x a y b x yxytu((t S a u S b) ((t = {x} u = {y}) x y)))
2817, 20brlec 6113 . . . 4 (ac bx a y b x y)
29 brco 4883 . . . . 5 (a(( S SI S ) S )bt(a S t t( S SI S )b))
30 brcnv 4892 . . . . . . . 8 (a S tt S a)
31 brco 4883 . . . . . . . . 9 (t( S SI S )bu(t SI S u u S b))
32 brsi 4761 . . . . . . . . . . . 12 (t SI S uxy(t = {x} u = {y} x S y))
33 df-3an 936 . . . . . . . . . . . . . 14 ((t = {x} u = {y} x S y) ↔ ((t = {x} u = {y}) x S y))
3416, 19brsset 4758 . . . . . . . . . . . . . . 15 (x S yx y)
3534anbi2i 675 . . . . . . . . . . . . . 14 (((t = {x} u = {y}) x S y) ↔ ((t = {x} u = {y}) x y))
3633, 35bitri 240 . . . . . . . . . . . . 13 ((t = {x} u = {y} x S y) ↔ ((t = {x} u = {y}) x y))
37362exbii 1583 . . . . . . . . . . . 12 (xy(t = {x} u = {y} x S y) ↔ xy((t = {x} u = {y}) x y))
3832, 37bitri 240 . . . . . . . . . . 11 (t SI S uxy((t = {x} u = {y}) x y))
3938anbi2ci 677 . . . . . . . . . 10 ((t SI S u u S b) ↔ (u S b xy((t = {x} u = {y}) x y)))
4039exbii 1582 . . . . . . . . 9 (u(t SI S u u S b) ↔ u(u S b xy((t = {x} u = {y}) x y)))
4131, 40bitri 240 . . . . . . . 8 (t( S SI S )bu(u S b xy((t = {x} u = {y}) x y)))
4230, 41anbi12i 678 . . . . . . 7 ((a S t t( S SI S )b) ↔ (t S a u(u S b xy((t = {x} u = {y}) x y))))
43 19.42v 1905 . . . . . . 7 (u(t S a (u S b xy((t = {x} u = {y}) x y))) ↔ (t S a u(u S b xy((t = {x} u = {y}) x y))))
44 19.42vv 1907 . . . . . . . . 9 (xy((t S a u S b) ((t = {x} u = {y}) x y)) ↔ ((t S a u S b) xy((t = {x} u = {y}) x y)))
45 anass 630 . . . . . . . . 9 (((t S a u S b) xy((t = {x} u = {y}) x y)) ↔ (t S a (u S b xy((t = {x} u = {y}) x y))))
4644, 45bitr2i 241 . . . . . . . 8 ((t S a (u S b xy((t = {x} u = {y}) x y))) ↔ xy((t S a u S b) ((t = {x} u = {y}) x y)))
4746exbii 1582 . . . . . . 7 (u(t S a (u S b xy((t = {x} u = {y}) x y))) ↔ uxy((t S a u S b) ((t = {x} u = {y}) x y)))
4842, 43, 473bitr2i 264 . . . . . 6 ((a S t t( S SI S )b) ↔ uxy((t S a u S b) ((t = {x} u = {y}) x y)))
4948exbii 1582 . . . . 5 (t(a S t t( S SI S )b) ↔ tuxy((t S a u S b) ((t = {x} u = {y}) x y)))
50 exrot4 1745 . . . . 5 (tuxy((t S a u S b) ((t = {x} u = {y}) x y)) ↔ xytu((t S a u S b) ((t = {x} u = {y}) x y)))
5129, 49, 503bitri 262 . . . 4 (a(( S SI S ) S )bxytu((t S a u S b) ((t = {x} u = {y}) x y)))
5227, 28, 513bitr4i 268 . . 3 (ac ba(( S SI S ) S )b)
5352eqbrriv 4851 . 2 c = (( S SI S ) S )
54 ssetex 4744 . . . 4 S V
5554siex 4753 . . . 4 SI S V
5654, 55coex 4750 . . 3 ( S SI S ) V
5754cnvex 5102 . . 3 S V
5856, 57coex 4750 . 2 (( S SI S ) S ) V
5953, 58eqeltri 2423 1 c V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  {csn 3737   class class class wbr 4639   S csset 4719   SI csi 4720   ∘ ccom 4721  ◡ccnv 4771   ≤c clec 6089 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-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-cnv 4785  df-lec 6099 This theorem is referenced by:  ltcex  6116  lecponc  6213  leconnnc  6218  nclennlem1  6248  nmembers1lem1  6268  nchoicelem4  6292
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