Theorem letsr 17050

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 9981	. . 3 ⊢ Rel ≤
2		lerelxr 9980	. . . . . . . . . . 11 ⊢ ≤ ⊆ (ℝ* × ℝ*)
3	2	brel 5090	. . . . . . . . . 10 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*))
4	3	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*))
5	4	simpld 474	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ∈ ℝ*)
6	4	simprd 478	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑦 ∈ ℝ*)
7	2	brel 5090	. . . . . . . . . 10 ⊢ (𝑦 ≤ 𝑧 → (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
8	7	simprd 478	. . . . . . . . 9 ⊢ (𝑦 ≤ 𝑧 → 𝑧 ∈ ℝ*)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑧 ∈ ℝ*)
10	5, 6, 9	3jca 1235	. . . . . . 7 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
11		xrletr 11865	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
12	10, 11	mpcom 37	. . . . . 6 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
13	12	ax-gen 1713	. . . . 5 ⊢ ∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
14	13	gen2 1714	. . . 4 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
15		cotr 5427	. . . 4 ⊢ (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
16	14, 15	mpbir 220	. . 3 ⊢ ( ≤ ∘ ≤ ) ⊆ ≤
17		asymref 5431	. . . 4 ⊢ (( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ) ↔ ∀𝑥 ∈ ∪ ∪ ≤ ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
18		simpr 476	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))
19	2	brel 5090	. . . . . . . . . . . 12 ⊢ (𝑦 ≤ 𝑥 → (𝑦 ∈ ℝ* ∧ 𝑥 ∈ ℝ*))
20	19	simpld 474	. . . . . . . . . . 11 ⊢ (𝑦 ≤ 𝑥 → 𝑦 ∈ ℝ*)
21	20	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑦 ∈ ℝ*)
22		xrletri3 11861	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
23	21, 22	sylan2 490	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
24	18, 23	mpbird 246	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑥 = 𝑦)
25	24	ex 449	. . . . . . 7 ⊢ (𝑥 ∈ ℝ* → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
26		xrleid 11859	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥)
27	26, 26	jca 553	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥))
28		breq2 4587	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑥 ≤ 𝑦))
29		breq1 4586	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑦 ≤ 𝑥))
30	28, 29	anbi12d 743	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥) ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
31	27, 30	syl5ibcom 234	. . . . . . 7 ⊢ (𝑥 ∈ ℝ* → (𝑥 = 𝑦 → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
32	25, 31	impbid 201	. . . . . 6 ⊢ (𝑥 ∈ ℝ* → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
33	32	alrimiv 1842	. . . . 5 ⊢ (𝑥 ∈ ℝ* → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
34		lefld 17049	. . . . . 6 ⊢ ℝ* = ∪ ∪ ≤
35	34	eqcomi 2619	. . . . 5 ⊢ ∪ ∪ ≤ = ℝ*
36	33, 35	eleq2s 2706	. . . 4 ⊢ (𝑥 ∈ ∪ ∪ ≤ → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
37	17, 36	mprgbir 2911	. . 3 ⊢ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )
38		xrex 11705	. . . . . 6 ⊢ ℝ* ∈ V
39	38, 38	xpex 6860	. . . . 5 ⊢ (ℝ* × ℝ*) ∈ V
40	39, 2	ssexi 4731	. . . 4 ⊢ ≤ ∈ V
41		isps 17025	. . . 4 ⊢ ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ))))
42	40, 41	ax-mp 5	. . 3 ⊢ ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )))
43	1, 16, 37, 42	mpbir3an 1237	. 2 ⊢ ≤ ∈ PosetRel
44		xrletri 11860	. . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
45	44	rgen2a 2960	. . 3 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)
46		qfto 5436	. . 3 ⊢ ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ ) ↔ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
47	45, 46	mpbir 220	. 2 ⊢ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )
48		ledm 17047	. . 3 ⊢ ℝ* = dom ≤
49	48	istsr 17040	. 2 ⊢ ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )))
50	43, 47, 49	mpbir2an 957	1 ⊢ ≤ ∈ TosetRel

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583   I cid 4948   × cxp 5036  ^◡ccnv 5037   ↾ cres 5040   ∘ ccom 5042  Rel wrel 5043  ℝ^*cxr 9952   ≤ cle 9954  PosetRelcps 17021   TosetRel ctsr 17022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-ps 17023  df-tsr 17024

This theorem is referenced by:  cnfldle  19576  letopon  20819  leordtval2  20826  leordtval  20827  iccordt  20828  ordtrestixx  20836  xrge0tsms  22445  icopnfhmeo  22550  iccpnfhmeo  22552  xrhmeo  22553  xrhaus  28925  xrge0tsmsd  29116  cnvordtrestixx  29287  xrmulc1cn  29304  xrge0iifhmeo  29310  poimir  32612