Theorem letsr 18550

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 11200	. . 3 ⊢ Rel ≤
2		lerelxr 11199	. . . . . . . . . . 11 ⊢ ≤ ⊆ (ℝ* × ℝ*)
3	2	brel 5689	. . . . . . . . . 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 5689	. . . . . . . . . 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 1129	. . . . . . 7 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
11		xrletr 13100	. . . . . . 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 1797	. . . . 5 ⊢ ∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
14	13	gen2 1798	. . . 4 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
15		cotr 6069	. . . 4 ⊢ (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
16	14, 15	mpbir 231	. . 3 ⊢ ( ≤ ∘ ≤ ) ⊆ ≤
17		asymref 6073	. . . 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 5689	. . . . . . . . . . . 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 13096	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
23	21, 22	sylan2 594	. . . . . . . . 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 13093	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥)
27	26, 26	jca 511	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥))
28		breq2 5090	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑥 ≤ 𝑦))
29		breq1 5089	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑦 ≤ 𝑥))
30	28, 29	anbi12d 633	. . . . . . . 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 1929	. . . . 5 ⊢ (𝑥 ∈ ℝ* → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
34		lefld 18549	. . . . . 6 ⊢ ℝ* = ∪ ∪ ≤
35	34	eqcomi 2746	. . . . 5 ⊢ ∪ ∪ ≤ = ℝ*
36	33, 35	eleq2s 2855	. . . 4 ⊢ (𝑥 ∈ ∪ ∪ ≤ → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
37	17, 36	mprgbir 3059	. . 3 ⊢ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )
38		xrex 12928	. . . . . 6 ⊢ ℝ* ∈ V
39	38, 38	xpex 7700	. . . . 5 ⊢ (ℝ* × ℝ*) ∈ V
40	39, 2	ssexi 5259	. . . 4 ⊢ ≤ ∈ V
41		isps 18525	. . . 4 ⊢ ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ))))
42	40, 41	ax-mp 5	. . 3 ⊢ ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )))
43	1, 16, 37, 42	mpbir3an 1343	. 2 ⊢ ≤ ∈ PosetRel
44		xrletri 13095	. . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
45	44	rgen2 3178	. . 3 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)
46		qfto 6078	. . 3 ⊢ ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ ) ↔ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
47	45, 46	mpbir 231	. 2 ⊢ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )
48		ledm 18547	. . 3 ⊢ ℝ* = dom ≤
49	48	istsr 18540	. 2 ⊢ ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )))
50	43, 47, 49	mpbir2an 712	1 ⊢ ≤ ∈ TosetRel

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∪ cuni 4851   class class class wbr 5086   I cid 5518   × cxp 5622  ^◡ccnv 5623   ↾ cres 5626   ∘ ccom 5628  Rel wrel 5629  ℝ^*cxr 11169   ≤ cle 11171  PosetRelcps 18521   TosetRel ctsr 18522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-ps 18523  df-tsr 18524

This theorem is referenced by:  cnfldle  21355  cnfldfun  21358  cnfldfunALT  21359  cnfldleOLD  21368  cnfldfunOLD  21371  cnfldfunALTOLD  21372  letopon  23180  leordtval2  23187  leordtval  23188  iccordt  23189  ordtrestixx  23197  xrhaus  23360  xrge0tsms  24810  icopnfhmeo  24920  iccpnfhmeo  24922  xrhmeo  24923  xrge0tsmsd  33149  cnvordtrestixx  34073  xrmulc1cn  34090  xrge0iifhmeo  34096  poimir  37988