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Theorem letsr 18487
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 11224 . . 3 Rel ≤
2 lerelxr 11223 . . . . . . . . . . 11 ≤ ⊆ (ℝ* × ℝ*)
32brel 5698 . . . . . . . . . 10 (𝑥𝑦 → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
43adantr 482 . . . . . . . . 9 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
54simpld 496 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑥 ∈ ℝ*)
64simprd 497 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑦 ∈ ℝ*)
72brel 5698 . . . . . . . . . 10 (𝑦𝑧 → (𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
87simprd 497 . . . . . . . . 9 (𝑦𝑧𝑧 ∈ ℝ*)
98adantl 483 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑧 ∈ ℝ*)
105, 6, 93jca 1129 . . . . . . 7 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
11 xrletr 13083 . . . . . . 7 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1210, 11mpcom 38 . . . . . 6 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1312ax-gen 1798 . . . . 5 𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1413gen2 1799 . . . 4 𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
15 cotr 6065 . . . 4 (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1614, 15mpbir 230 . . 3 ( ≤ ∘ ≤ ) ⊆ ≤
17 asymref 6071 . . . 4 (( ≤ ∩ ≤ ) = ( I ↾ ≤ ) ↔ ∀𝑥 ≤ ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
18 simpr 486 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥𝑦𝑦𝑥))
192brel 5698 . . . . . . . . . . . 12 (𝑦𝑥 → (𝑦 ∈ ℝ*𝑥 ∈ ℝ*))
2019simpld 496 . . . . . . . . . . 11 (𝑦𝑥𝑦 ∈ ℝ*)
2120adantl 483 . . . . . . . . . 10 ((𝑥𝑦𝑦𝑥) → 𝑦 ∈ ℝ*)
22 xrletri3 13079 . . . . . . . . . 10 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2321, 22sylan2 594 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2418, 23mpbird 257 . . . . . . . 8 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → 𝑥 = 𝑦)
2524ex 414 . . . . . . 7 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
26 xrleid 13076 . . . . . . . . 9 (𝑥 ∈ ℝ*𝑥𝑥)
2726, 26jca 513 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑥𝑥𝑥𝑥))
28 breq2 5110 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
29 breq1 5109 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
3028, 29anbi12d 632 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝑥𝑥𝑥) ↔ (𝑥𝑦𝑦𝑥)))
3127, 30syl5ibcom 244 . . . . . . 7 (𝑥 ∈ ℝ* → (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥)))
3225, 31impbid 211 . . . . . 6 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3332alrimiv 1931 . . . . 5 (𝑥 ∈ ℝ* → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
34 lefld 18486 . . . . . 6 * =
3534eqcomi 2742 . . . . 5 ≤ = ℝ*
3633, 35eleq2s 2852 . . . 4 (𝑥 ≤ → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3717, 36mprgbir 3068 . . 3 ( ≤ ∩ ≤ ) = ( I ↾ ≤ )
38 xrex 12917 . . . . . 6 * ∈ V
3938, 38xpex 7688 . . . . 5 (ℝ* × ℝ*) ∈ V
4039, 2ssexi 5280 . . . 4 ≤ ∈ V
41 isps 18462 . . . 4 ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ ))))
4240, 41ax-mp 5 . . 3 ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ )))
431, 16, 37, 42mpbir3an 1342 . 2 ≤ ∈ PosetRel
44 xrletri 13078 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥𝑦𝑦𝑥))
4544rgen2 3191 . . 3 𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥)
46 qfto 6076 . . 3 ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ ) ↔ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥))
4745, 46mpbir 230 . 2 (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )
48 ledm 18484 . . 3 * = dom ≤
4948istsr 18477 . 2 ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )))
5043, 47, 49mpbir2an 710 1 ≤ ∈ TosetRel
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088  wal 1540   = wceq 1542  wcel 2107  wral 3061  Vcvv 3444  cun 3909  cin 3910  wss 3911   cuni 4866   class class class wbr 5106   I cid 5531   × cxp 5632  ccnv 5633  cres 5636  ccom 5638  Rel wrel 5639  *cxr 11193  cle 11195  PosetRelcps 18458   TosetRel ctsr 18459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-pre-lttri 11130  ax-pre-lttrn 11131
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-ps 18460  df-tsr 18461
This theorem is referenced by:  cnfldle  20821  cnfldfun  20824  cnfldfunALT  20825  cnfldfunALTOLD  20826  letopon  22572  leordtval2  22579  leordtval  22580  iccordt  22581  ordtrestixx  22589  xrhaus  22752  xrge0tsms  24213  icopnfhmeo  24322  iccpnfhmeo  24324  xrhmeo  24325  xrge0tsmsd  31948  cnvordtrestixx  32551  xrmulc1cn  32568  xrge0iifhmeo  32574  poimir  36157
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