Theorem letsr 18618

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 11328	. . 3 ⊢ Rel ≤
2		lerelxr 11327	. . . . . . . . . . 11 ⊢ ≤ ⊆ (ℝ* × ℝ*)
3	2	brel 5747	. . . . . . . . . 10 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*))
4	3	adantr 479	. . . . . . . . 9 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*))
5	4	simpld 493	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ∈ ℝ*)
6	4	simprd 494	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑦 ∈ ℝ*)
7	2	brel 5747	. . . . . . . . . 10 ⊢ (𝑦 ≤ 𝑧 → (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
8	7	simprd 494	. . . . . . . . 9 ⊢ (𝑦 ≤ 𝑧 → 𝑧 ∈ ℝ*)
9	8	adantl 480	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑧 ∈ ℝ*)
10	5, 6, 9	3jca 1125	. . . . . . 7 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
11		xrletr 13191	. . . . . . 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 1790	. . . . 5 ⊢ ∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
14	13	gen2 1791	. . . 4 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
15		cotr 6122	. . . 4 ⊢ (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
16	14, 15	mpbir 230	. . 3 ⊢ ( ≤ ∘ ≤ ) ⊆ ≤
17		asymref 6128	. . . 4 ⊢ (( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ) ↔ ∀𝑥 ∈ ∪ ∪ ≤ ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
18		simpr 483	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))
19	2	brel 5747	. . . . . . . . . . . 12 ⊢ (𝑦 ≤ 𝑥 → (𝑦 ∈ ℝ* ∧ 𝑥 ∈ ℝ*))
20	19	simpld 493	. . . . . . . . . . 11 ⊢ (𝑦 ≤ 𝑥 → 𝑦 ∈ ℝ*)
21	20	adantl 480	. . . . . . . . . 10 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑦 ∈ ℝ*)
22		xrletri3 13187	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
23	21, 22	sylan2 591	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
24	18, 23	mpbird 256	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑥 = 𝑦)
25	24	ex 411	. . . . . . 7 ⊢ (𝑥 ∈ ℝ* → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
26		xrleid 13184	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥)
27	26, 26	jca 510	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥))
28		breq2 5157	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑥 ≤ 𝑦))
29		breq1 5156	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑦 ≤ 𝑥))
30	28, 29	anbi12d 630	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥) ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
31	27, 30	syl5ibcom 244	. . . . . . 7 ⊢ (𝑥 ∈ ℝ* → (𝑥 = 𝑦 → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
32	25, 31	impbid 211	. . . . . 6 ⊢ (𝑥 ∈ ℝ* → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
33	32	alrimiv 1923	. . . . 5 ⊢ (𝑥 ∈ ℝ* → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
34		lefld 18617	. . . . . 6 ⊢ ℝ* = ∪ ∪ ≤
35	34	eqcomi 2735	. . . . 5 ⊢ ∪ ∪ ≤ = ℝ*
36	33, 35	eleq2s 2844	. . . 4 ⊢ (𝑥 ∈ ∪ ∪ ≤ → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
37	17, 36	mprgbir 3058	. . 3 ⊢ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )
38		xrex 13023	. . . . . 6 ⊢ ℝ* ∈ V
39	38, 38	xpex 7761	. . . . 5 ⊢ (ℝ* × ℝ*) ∈ V
40	39, 2	ssexi 5327	. . . 4 ⊢ ≤ ∈ V
41		isps 18593	. . . 4 ⊢ ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ))))
42	40, 41	ax-mp 5	. . 3 ⊢ ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )))
43	1, 16, 37, 42	mpbir3an 1338	. 2 ⊢ ≤ ∈ PosetRel
44		xrletri 13186	. . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
45	44	rgen2 3188	. . 3 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)
46		qfto 6133	. . 3 ⊢ ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ ) ↔ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
47	45, 46	mpbir 230	. 2 ⊢ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )
48		ledm 18615	. . 3 ⊢ ℝ* = dom ≤
49	48	istsr 18608	. 2 ⊢ ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )))
50	43, 47, 49	mpbir2an 709	1 ⊢ ≤ ∈ TosetRel

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∀wral 3051  Vcvv 3462   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∪ cuni 4913   class class class wbr 5153   I cid 5579   × cxp 5680  ^◡ccnv 5681   ↾ cres 5684   ∘ ccom 5686  Rel wrel 5687  ℝ^*cxr 11297   ≤ cle 11299  PosetRelcps 18589   TosetRel ctsr 18590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-pre-lttri 11232  ax-pre-lttrn 11233

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-ps 18591  df-tsr 18592

This theorem is referenced by:  cnfldle  21354  cnfldfun  21357  cnfldfunALT  21358  cnfldleOLD  21367  cnfldfunOLD  21370  cnfldfunALTOLD  21371  cnfldfunALTOLDOLD  21372  letopon  23200  leordtval2  23207  leordtval  23208  iccordt  23209  ordtrestixx  23217  xrhaus  23380  xrge0tsms  24841  icopnfhmeo  24959  iccpnfhmeo  24961  xrhmeo  24962  xrge0tsmsd  32926  cnvordtrestixx  33728  xrmulc1cn  33745  xrge0iifhmeo  33751  poimir  37354