Theorem letsr 18496

Description: The "less than or equal to" relationship on the extended reals is a toset. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
letsr	⊢ ≤ ∈ TosetRel

Proof of Theorem letsr

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lerel 11173	. . 3 ⊢ Rel ≤
2		lerelxr 11172	. . . . . . . . . . 11 ⊢ ≤ ⊆ (ℝ* × ℝ*)
3	2	brel 5681	. . . . . . . . . 10 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*))
4	3	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*))
5	4	simpld 494	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ∈ ℝ*)
6	4	simprd 495	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑦 ∈ ℝ*)
7	2	brel 5681	. . . . . . . . . 10 ⊢ (𝑦 ≤ 𝑧 → (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
8	7	simprd 495	. . . . . . . . 9 ⊢ (𝑦 ≤ 𝑧 → 𝑧 ∈ ℝ*)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑧 ∈ ℝ*)
10	5, 6, 9	3jca 1128	. . . . . . 7 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
11		xrletr 13054	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
12	10, 11	mpcom 38	. . . . . 6 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
13	12	ax-gen 1796	. . . . 5 ⊢ ∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
14	13	gen2 1797	. . . 4 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
15		cotr 6059	. . . 4 ⊢ (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
16	14, 15	mpbir 231	. . 3 ⊢ ( ≤ ∘ ≤ ) ⊆ ≤
17		asymref 6063	. . . 4 ⊢ (( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ) ↔ ∀𝑥 ∈ ∪ ∪ ≤ ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
18		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))
19	2	brel 5681	. . . . . . . . . . . 12 ⊢ (𝑦 ≤ 𝑥 → (𝑦 ∈ ℝ* ∧ 𝑥 ∈ ℝ*))
20	19	simpld 494	. . . . . . . . . . 11 ⊢ (𝑦 ≤ 𝑥 → 𝑦 ∈ ℝ*)
21	20	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑦 ∈ ℝ*)
22		xrletri3 13050	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
23	21, 22	sylan2 593	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
24	18, 23	mpbird 257	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑥 = 𝑦)
25	24	ex 412	. . . . . . 7 ⊢ (𝑥 ∈ ℝ* → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
26		xrleid 13047	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥)
27	26, 26	jca 511	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥))
28		breq2 5095	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑥 ≤ 𝑦))
29		breq1 5094	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑦 ≤ 𝑥))
30	28, 29	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥) ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
31	27, 30	syl5ibcom 245	. . . . . . 7 ⊢ (𝑥 ∈ ℝ* → (𝑥 = 𝑦 → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
32	25, 31	impbid 212	. . . . . 6 ⊢ (𝑥 ∈ ℝ* → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
33	32	alrimiv 1928	. . . . 5 ⊢ (𝑥 ∈ ℝ* → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
34		lefld 18495	. . . . . 6 ⊢ ℝ* = ∪ ∪ ≤
35	34	eqcomi 2740	. . . . 5 ⊢ ∪ ∪ ≤ = ℝ*
36	33, 35	eleq2s 2849	. . . 4 ⊢ (𝑥 ∈ ∪ ∪ ≤ → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
37	17, 36	mprgbir 3054	. . 3 ⊢ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )
38		xrex 12882	. . . . . 6 ⊢ ℝ* ∈ V
39	38, 38	xpex 7686	. . . . 5 ⊢ (ℝ* × ℝ*) ∈ V
40	39, 2	ssexi 5260	. . . 4 ⊢ ≤ ∈ V
41		isps 18471	. . . 4 ⊢ ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ))))
42	40, 41	ax-mp 5	. . 3 ⊢ ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )))
43	1, 16, 37, 42	mpbir3an 1342	. 2 ⊢ ≤ ∈ PosetRel
44		xrletri 13049	. . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
45	44	rgen2 3172	. . 3 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)
46		qfto 6068	. . 3 ⊢ ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ ) ↔ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
47	45, 46	mpbir 231	. 2 ⊢ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )
48		ledm 18493	. . 3 ⊢ ℝ* = dom ≤
49	48	istsr 18486	. 2 ⊢ ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )))
50	43, 47, 49	mpbir2an 711	1 ⊢ ≤ ∈ TosetRel

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4859   class class class wbr 5091   I cid 5510   × cxp 5614  ^◡ccnv 5615   ↾ cres 5618   ∘ ccom 5620  Rel wrel 5621  ℝ^*cxr 11142   ≤ cle 11144  PosetRelcps 18467   TosetRel ctsr 18468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-pre-lttri 11077  ax-pre-lttrn 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-po 5524  df-so 5525  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-ps 18469  df-tsr 18470

This theorem is referenced by:  cnfldle  21300  cnfldfun  21303  cnfldfunALT  21304  cnfldleOLD  21313  cnfldfunOLD  21316  cnfldfunALTOLD  21317  letopon  23118  leordtval2  23125  leordtval  23126  iccordt  23127  ordtrestixx  23135  xrhaus  23298  xrge0tsms  24748  icopnfhmeo  24866  iccpnfhmeo  24868  xrhmeo  24869  xrge0tsmsd  33037  cnvordtrestixx  33921  xrmulc1cn  33938  xrge0iifhmeo  33944  poimir  37692