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Theorem letsr 17708
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 10504 . . 3 Rel ≤
2 lerelxr 10503 . . . . . . . . . . 11 ≤ ⊆ (ℝ* × ℝ*)
32brel 5464 . . . . . . . . . 10 (𝑥𝑦 → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
43adantr 473 . . . . . . . . 9 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
54simpld 487 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑥 ∈ ℝ*)
64simprd 488 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑦 ∈ ℝ*)
72brel 5464 . . . . . . . . . 10 (𝑦𝑧 → (𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
87simprd 488 . . . . . . . . 9 (𝑦𝑧𝑧 ∈ ℝ*)
98adantl 474 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑧 ∈ ℝ*)
105, 6, 93jca 1109 . . . . . . 7 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
11 xrletr 12367 . . . . . . 7 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1210, 11mpcom 38 . . . . . 6 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1312ax-gen 1759 . . . . 5 𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1413gen2 1760 . . . 4 𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
15 cotr 5810 . . . 4 (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1614, 15mpbir 223 . . 3 ( ≤ ∘ ≤ ) ⊆ ≤
17 asymref 5814 . . . 4 (( ≤ ∩ ≤ ) = ( I ↾ ≤ ) ↔ ∀𝑥 ≤ ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
18 simpr 477 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥𝑦𝑦𝑥))
192brel 5464 . . . . . . . . . . . 12 (𝑦𝑥 → (𝑦 ∈ ℝ*𝑥 ∈ ℝ*))
2019simpld 487 . . . . . . . . . . 11 (𝑦𝑥𝑦 ∈ ℝ*)
2120adantl 474 . . . . . . . . . 10 ((𝑥𝑦𝑦𝑥) → 𝑦 ∈ ℝ*)
22 xrletri3 12363 . . . . . . . . . 10 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2321, 22sylan2 584 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2418, 23mpbird 249 . . . . . . . 8 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → 𝑥 = 𝑦)
2524ex 405 . . . . . . 7 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
26 xrleid 12360 . . . . . . . . 9 (𝑥 ∈ ℝ*𝑥𝑥)
2726, 26jca 504 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑥𝑥𝑥𝑥))
28 breq2 4930 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
29 breq1 4929 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
3028, 29anbi12d 622 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝑥𝑥𝑥) ↔ (𝑥𝑦𝑦𝑥)))
3127, 30syl5ibcom 237 . . . . . . 7 (𝑥 ∈ ℝ* → (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥)))
3225, 31impbid 204 . . . . . 6 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3332alrimiv 1887 . . . . 5 (𝑥 ∈ ℝ* → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
34 lefld 17707 . . . . . 6 * =
3534eqcomi 2782 . . . . 5 ≤ = ℝ*
3633, 35eleq2s 2879 . . . 4 (𝑥 ≤ → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3717, 36mprgbir 3098 . . 3 ( ≤ ∩ ≤ ) = ( I ↾ ≤ )
38 xrex 12200 . . . . . 6 * ∈ V
3938, 38xpex 7292 . . . . 5 (ℝ* × ℝ*) ∈ V
4039, 2ssexi 5079 . . . 4 ≤ ∈ V
41 isps 17683 . . . 4 ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ ))))
4240, 41ax-mp 5 . . 3 ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ )))
431, 16, 37, 42mpbir3an 1322 . 2 ≤ ∈ PosetRel
44 xrletri 12362 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥𝑦𝑦𝑥))
4544rgen2a 3171 . . 3 𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥)
46 qfto 5819 . . 3 ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ ) ↔ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥))
4745, 46mpbir 223 . 2 (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )
48 ledm 17705 . . 3 * = dom ≤
4948istsr 17698 . 2 ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )))
5043, 47, 49mpbir2an 699 1 ≤ ∈ TosetRel
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wo 834  w3a 1069  wal 1506   = wceq 1508  wcel 2051  wral 3083  Vcvv 3410  cun 3822  cin 3823  wss 3824   cuni 4709   class class class wbr 4926   I cid 5308   × cxp 5402  ccnv 5403  cres 5406  ccom 5408  Rel wrel 5409  *cxr 10472  cle 10474  PosetRelcps 17679   TosetRel ctsr 17680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-pre-lttri 10408  ax-pre-lttrn 10409
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-po 5323  df-so 5324  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-er 8088  df-en 8306  df-dom 8307  df-sdom 8308  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-ps 17681  df-tsr 17682
This theorem is referenced by:  cnfldle  20272  cnfldfun  20275  cnfldfunALT  20276  letopon  21533  leordtval2  21540  leordtval  21541  iccordt  21542  ordtrestixx  21550  xrhaus  21713  xrge0tsms  23161  icopnfhmeo  23266  iccpnfhmeo  23268  xrhmeo  23269  xrge0tsmsd  30563  cnvordtrestixx  30833  xrmulc1cn  30850  xrge0iifhmeo  30856  poimir  34399
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