Theorem lecex 6116

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.

Step	Hyp	Ref	Expression
1		r2ex 2653	. . . . 5 ⊢ (∃x ∈ a ∃y ∈ b x ⊆ y ↔ ∃x∃y((x ∈ a ∧ y ∈ b) ∧ x ⊆ y))
2		19.41vv 1902	. . . . . . 7 ⊢ (∃t∃u(((t S a ∧ u S b) ∧ (t = {x} ∧ u = {y})) ∧ x ⊆ y) ↔ (∃t∃u((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)))
4	3	2exbii 1583	. . . . . . 7 ⊢ (∃t∃u(((t S a ∧ u S b) ∧ (t = {x} ∧ u = {y})) ∧ x ⊆ y) ↔ ∃t∃u((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)))
7	5, 6	bitr4i 243	. . . . . . . . . 10 ⊢ (((t S a ∧ u S b) ∧ (t = {x} ∧ u = {y})) ↔ (t = {x} ∧ u = {y} ∧ (t S a ∧ u S b)))
8	7	2exbii 1583	. . . . . . . . 9 ⊢ (∃t∃u((t S a ∧ u S b) ∧ (t = {x} ∧ u = {y})) ↔ ∃t∃u(t = {x} ∧ u = {y} ∧ (t S a ∧ u S b)))
9		snex 4112	. . . . . . . . . 10 ⊢ {x} ∈ V
10		snex 4112	. . . . . . . . . 10 ⊢ {y} ∈ V
11		breq1 4643	. . . . . . . . . . 11 ⊢ (t = {x} → (t S a ↔ {x} S a))
12	11	anbi1d 685	. . . . . . . . . 10 ⊢ (t = {x} → ((t S a ∧ u S b) ↔ ({x} S a ∧ u S b)))
13		breq1 4643	. . . . . . . . . . 11 ⊢ (u = {y} → (u S b ↔ {y} S b))
14	13	anbi2d 684	. . . . . . . . . 10 ⊢ (u = {y} → (({x} S a ∧ u S b) ↔ ({x} S a ∧ {y} S b)))
15	9, 10, 12, 14	ceqsex2v 2897	. . . . . . . . 9 ⊢ (∃t∃u(t = {x} ∧ u = {y} ∧ (t S a ∧ u S b)) ↔ ({x} S a ∧ {y} S b))
16		vex 2863	. . . . . . . . . . 11 ⊢ x ∈ V
17		vex 2863	. . . . . . . . . . 11 ⊢ a ∈ V
18	16, 17	brssetsn 4760	. . . . . . . . . 10 ⊢ ({x} S a ↔ x ∈ a)
19		vex 2863	. . . . . . . . . . 11 ⊢ y ∈ V
20		vex 2863	. . . . . . . . . . 11 ⊢ b ∈ V
21	19, 20	brssetsn 4760	. . . . . . . . . 10 ⊢ ({y} S b ↔ y ∈ b)
22	18, 21	anbi12i 678	. . . . . . . . 9 ⊢ (({x} S a ∧ {y} S b) ↔ (x ∈ a ∧ y ∈ b))
23	8, 15, 22	3bitri 262	. . . . . . . 8 ⊢ (∃t∃u((t S a ∧ u S b) ∧ (t = {x} ∧ u = {y})) ↔ (x ∈ a ∧ y ∈ b))
24	23	anbi1i 676	. . . . . . 7 ⊢ ((∃t∃u((t S a ∧ u S b) ∧ (t = {x} ∧ u = {y})) ∧ x ⊆ y) ↔ ((x ∈ a ∧ y ∈ b) ∧ x ⊆ y))
25	2, 4, 24	3bitr3i 266	. . . . . 6 ⊢ (∃t∃u((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)) ↔ ((x ∈ a ∧ y ∈ b) ∧ x ⊆ y))
26	25	2exbii 1583	. . . . 5 ⊢ (∃x∃y∃t∃u((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)) ↔ ∃x∃y((x ∈ a ∧ y ∈ b) ∧ x ⊆ y))
27	1, 26	bitr4i 243	. . . 4 ⊢ (∃x ∈ a ∃y ∈ b x ⊆ y ↔ ∃x∃y∃t∃u((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
28	17, 20	brlec 6114	. . . 4 ⊢ (a ≤c b ↔ ∃x ∈ a ∃y ∈ b x ⊆ y)
29		brco 4884	. . . . 5 ⊢ (a(( S ∘ SI S ) ∘ ◡ S )b ↔ ∃t(a◡ S t ∧ t( S ∘ SI S )b))
30		brcnv 4893	. . . . . . . 8 ⊢ (a◡ S t ↔ t S a)
31		brco 4884	. . . . . . . . 9 ⊢ (t( S ∘ SI S )b ↔ ∃u(t SI S u ∧ u S b))
32		brsi 4762	. . . . . . . . . . . 12 ⊢ (t SI S u ↔ ∃x∃y(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))
34	16, 19	brsset 4759	. . . . . . . . . . . . . . 15 ⊢ (x S y ↔ x ⊆ y)
35	34	anbi2i 675	. . . . . . . . . . . . . 14 ⊢ (((t = {x} ∧ u = {y}) ∧ x S y) ↔ ((t = {x} ∧ u = {y}) ∧ x ⊆ y))
36	33, 35	bitri 240	. . . . . . . . . . . . 13 ⊢ ((t = {x} ∧ u = {y} ∧ x S y) ↔ ((t = {x} ∧ u = {y}) ∧ x ⊆ y))
37	36	2exbii 1583	. . . . . . . . . . . 12 ⊢ (∃x∃y(t = {x} ∧ u = {y} ∧ x S y) ↔ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y))
38	32, 37	bitri 240	. . . . . . . . . . 11 ⊢ (t SI S u ↔ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y))
39	38	anbi2ci 677	. . . . . . . . . 10 ⊢ ((t SI S u ∧ u S b) ↔ (u S b ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
40	39	exbii 1582	. . . . . . . . 9 ⊢ (∃u(t SI S u ∧ u S b) ↔ ∃u(u S b ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
41	31, 40	bitri 240	. . . . . . . 8 ⊢ (t( S ∘ SI S )b ↔ ∃u(u S b ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
42	30, 41	anbi12i 678	. . . . . . 7 ⊢ ((a◡ S t ∧ t( S ∘ SI S )b) ↔ (t S a ∧ ∃u(u S b ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y))))
43		19.42v 1905	. . . . . . 7 ⊢ (∃u(t S a ∧ (u S b ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y))) ↔ (t S a ∧ ∃u(u S b ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y))))
44		19.42vv 1907	. . . . . . . . 9 ⊢ (∃x∃y((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)) ↔ ((t S a ∧ u S b) ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
45		anass 630	. . . . . . . . 9 ⊢ (((t S a ∧ u S b) ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y)) ↔ (t S a ∧ (u S b ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y))))
46	44, 45	bitr2i 241	. . . . . . . 8 ⊢ ((t S a ∧ (u S b ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y))) ↔ ∃x∃y((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
47	46	exbii 1582	. . . . . . 7 ⊢ (∃u(t S a ∧ (u S b ∧ ∃x∃y((t = {x} ∧ u = {y}) ∧ x ⊆ y))) ↔ ∃u∃x∃y((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
48	42, 43, 47	3bitr2i 264	. . . . . 6 ⊢ ((a◡ S t ∧ t( S ∘ SI S )b) ↔ ∃u∃x∃y((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
49	48	exbii 1582	. . . . 5 ⊢ (∃t(a◡ S t ∧ t( S ∘ SI S )b) ↔ ∃t∃u∃x∃y((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
50		exrot4 1745	. . . . 5 ⊢ (∃t∃u∃x∃y((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)) ↔ ∃x∃y∃t∃u((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
51	29, 49, 50	3bitri 262	. . . 4 ⊢ (a(( S ∘ SI S ) ∘ ◡ S )b ↔ ∃x∃y∃t∃u((t S a ∧ u S b) ∧ ((t = {x} ∧ u = {y}) ∧ x ⊆ y)))
52	27, 28, 51	3bitr4i 268	. . 3 ⊢ (a ≤c b ↔ a(( S ∘ SI S ) ∘ ◡ S )b)
53	52	eqbrriv 4852	. 2 ⊢ ≤c = (( S ∘ SI S ) ∘ ◡ S )
54		ssetex 4745	. . . 4 ⊢ S ∈ V
55	54	siex 4754	. . . 4 ⊢ SI S ∈ V
56	54, 55	coex 4751	. . 3 ⊢ ( S ∘ SI S ) ∈ V
57	54	cnvex 5103	. . 3 ⊢ ◡ S ∈ V
58	56, 57	coex 4751	. 2 ⊢ (( S ∘ SI S ) ∘ ◡ S ) ∈ V
59	53, 58	eqeltri 2423	1 ⊢ ≤c ∈ V

This proof depends on syntax axioms:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258  {csn 3738   class class class wbr 4640   S csset 4720   SI csi 4721   ∘ ccom 4722  ^◡ccnv 4772   ≤_c clec 6090

This proof depends on 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 proof 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-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-cnv 4786  df-lec 6100

This theorem is used by:  ltcex  6117  lecponc  6214  leconnnc  6219  nclennlem1  6249  nmembers1lem1  6269  nchoicelem4  6293