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Theorem letsr 18651
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.
StepHypRef Expression
1 lerel 11323 . . 3 Rel ≤
2 lerelxr 11322 . . . . . . . . . . 11 ≤ ⊆ (ℝ* × ℝ*)
32brel 5754 . . . . . . . . . 10 (𝑥𝑦 → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
43adantr 480 . . . . . . . . 9 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
54simpld 494 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑥 ∈ ℝ*)
64simprd 495 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑦 ∈ ℝ*)
72brel 5754 . . . . . . . . . 10 (𝑦𝑧 → (𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
87simprd 495 . . . . . . . . 9 (𝑦𝑧𝑧 ∈ ℝ*)
98adantl 481 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑧 ∈ ℝ*)
105, 6, 93jca 1127 . . . . . . 7 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
11 xrletr 13197 . . . . . . 7 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1210, 11mpcom 38 . . . . . 6 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1312ax-gen 1792 . . . . 5 𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1413gen2 1793 . . . 4 𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
15 cotr 6133 . . . 4 (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1614, 15mpbir 231 . . 3 ( ≤ ∘ ≤ ) ⊆ ≤
17 asymref 6139 . . . 4 (( ≤ ∩ ≤ ) = ( I ↾ ≤ ) ↔ ∀𝑥 ≤ ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
18 simpr 484 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥𝑦𝑦𝑥))
192brel 5754 . . . . . . . . . . . 12 (𝑦𝑥 → (𝑦 ∈ ℝ*𝑥 ∈ ℝ*))
2019simpld 494 . . . . . . . . . . 11 (𝑦𝑥𝑦 ∈ ℝ*)
2120adantl 481 . . . . . . . . . 10 ((𝑥𝑦𝑦𝑥) → 𝑦 ∈ ℝ*)
22 xrletri3 13193 . . . . . . . . . 10 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2321, 22sylan2 593 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2418, 23mpbird 257 . . . . . . . 8 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → 𝑥 = 𝑦)
2524ex 412 . . . . . . 7 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
26 xrleid 13190 . . . . . . . . 9 (𝑥 ∈ ℝ*𝑥𝑥)
2726, 26jca 511 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑥𝑥𝑥𝑥))
28 breq2 5152 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
29 breq1 5151 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
3028, 29anbi12d 632 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝑥𝑥𝑥) ↔ (𝑥𝑦𝑦𝑥)))
3127, 30syl5ibcom 245 . . . . . . 7 (𝑥 ∈ ℝ* → (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥)))
3225, 31impbid 212 . . . . . 6 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3332alrimiv 1925 . . . . 5 (𝑥 ∈ ℝ* → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
34 lefld 18650 . . . . . 6 * =
3534eqcomi 2744 . . . . 5 ≤ = ℝ*
3633, 35eleq2s 2857 . . . 4 (𝑥 ≤ → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3717, 36mprgbir 3066 . . 3 ( ≤ ∩ ≤ ) = ( I ↾ ≤ )
38 xrex 13027 . . . . . 6 * ∈ V
3938, 38xpex 7772 . . . . 5 (ℝ* × ℝ*) ∈ V
4039, 2ssexi 5328 . . . 4 ≤ ∈ V
41 isps 18626 . . . 4 ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ ))))
4240, 41ax-mp 5 . . 3 ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ )))
431, 16, 37, 42mpbir3an 1340 . 2 ≤ ∈ PosetRel
44 xrletri 13192 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥𝑦𝑦𝑥))
4544rgen2 3197 . . 3 𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥)
46 qfto 6144 . . 3 ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ ) ↔ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥))
4745, 46mpbir 231 . 2 (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )
48 ledm 18648 . . 3 * = dom ≤
4948istsr 18641 . 2 ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )))
5043, 47, 49mpbir2an 711 1 ≤ ∈ TosetRel
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cun 3961  cin 3962  wss 3963   cuni 4912   class class class wbr 5148   I cid 5582   × cxp 5687  ccnv 5688  cres 5691  ccom 5693  Rel wrel 5694  *cxr 11292  cle 11294  PosetRelcps 18622   TosetRel ctsr 18623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-ps 18624  df-tsr 18625
This theorem is referenced by:  cnfldle  21393  cnfldfun  21396  cnfldfunALT  21397  cnfldleOLD  21406  cnfldfunOLD  21409  cnfldfunALTOLD  21410  cnfldfunALTOLDOLD  21411  letopon  23229  leordtval2  23236  leordtval  23237  iccordt  23238  ordtrestixx  23246  xrhaus  23409  xrge0tsms  24870  icopnfhmeo  24988  iccpnfhmeo  24990  xrhmeo  24991  xrge0tsmsd  33048  cnvordtrestixx  33874  xrmulc1cn  33891  xrge0iifhmeo  33897  poimir  37640
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