Theorem letsr 17435

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 10304	. . 3 ⊢ Rel ≤
2		lerelxr 10303	. . . . . . . . . . 11 ⊢ ≤ ⊆ (ℝ* × ℝ*)
3	2	brel 5308	. . . . . . . . . 10 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*))
4	3	adantr 466	. . . . . . . . 9 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*))
5	4	simpld 476	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ∈ ℝ*)
6	4	simprd 477	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑦 ∈ ℝ*)
7	2	brel 5308	. . . . . . . . . 10 ⊢ (𝑦 ≤ 𝑧 → (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
8	7	simprd 477	. . . . . . . . 9 ⊢ (𝑦 ≤ 𝑧 → 𝑧 ∈ ℝ*)
9	8	adantl 467	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑧 ∈ ℝ*)
10	5, 6, 9	3jca 1122	. . . . . . 7 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
11		xrletr 12194	. . . . . . 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 1870	. . . . 5 ⊢ ∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
14	13	gen2 1871	. . . 4 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)
15		cotr 5649	. . . 4 ⊢ (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
16	14, 15	mpbir 221	. . 3 ⊢ ( ≤ ∘ ≤ ) ⊆ ≤
17		asymref 5653	. . . 4 ⊢ (( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ) ↔ ∀𝑥 ∈ ∪ ∪ ≤ ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
18		simpr 471	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))
19	2	brel 5308	. . . . . . . . . . . 12 ⊢ (𝑦 ≤ 𝑥 → (𝑦 ∈ ℝ* ∧ 𝑥 ∈ ℝ*))
20	19	simpld 476	. . . . . . . . . . 11 ⊢ (𝑦 ≤ 𝑥 → 𝑦 ∈ ℝ*)
21	20	adantl 467	. . . . . . . . . 10 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑦 ∈ ℝ*)
22		xrletri3 12190	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
23	21, 22	sylan2 572	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
24	18, 23	mpbird 247	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑥 = 𝑦)
25	24	ex 397	. . . . . . 7 ⊢ (𝑥 ∈ ℝ* → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
26		xrleid 12188	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥)
27	26, 26	jca 495	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥))
28		breq2 4790	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑥 ≤ 𝑦))
29		breq1 4789	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑦 ≤ 𝑥))
30	28, 29	anbi12d 608	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥) ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
31	27, 30	syl5ibcom 235	. . . . . . 7 ⊢ (𝑥 ∈ ℝ* → (𝑥 = 𝑦 → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
32	25, 31	impbid 202	. . . . . 6 ⊢ (𝑥 ∈ ℝ* → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
33	32	alrimiv 2007	. . . . 5 ⊢ (𝑥 ∈ ℝ* → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
34		lefld 17434	. . . . . 6 ⊢ ℝ* = ∪ ∪ ≤
35	34	eqcomi 2780	. . . . 5 ⊢ ∪ ∪ ≤ = ℝ*
36	33, 35	eleq2s 2868	. . . 4 ⊢ (𝑥 ∈ ∪ ∪ ≤ → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦))
37	17, 36	mprgbir 3076	. . 3 ⊢ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )
38		xrex 12032	. . . . . 6 ⊢ ℝ* ∈ V
39	38, 38	xpex 7109	. . . . 5 ⊢ (ℝ* × ℝ*) ∈ V
40	39, 2	ssexi 4937	. . . 4 ⊢ ≤ ∈ V
41		isps 17410	. . . 4 ⊢ ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ))))
42	40, 41	ax-mp 5	. . 3 ⊢ ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )))
43	1, 16, 37, 42	mpbir3an 1426	. 2 ⊢ ≤ ∈ PosetRel
44		xrletri 12189	. . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
45	44	rgen2a 3126	. . 3 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)
46		qfto 5658	. . 3 ⊢ ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ ) ↔ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
47	45, 46	mpbir 221	. 2 ⊢ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )
48		ledm 17432	. . 3 ⊢ ℝ* = dom ≤
49	48	istsr 17425	. 2 ⊢ ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ◡ ≤ )))
50	43, 47, 49	mpbir2an 682	1 ⊢ ≤ ∈ TosetRel

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 826   ∧ w3a 1071  ∀wal 1629   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ∪ cun 3721   ∩ cin 3722   ⊆ wss 3723  ∪ cuni 4574   class class class wbr 4786   I cid 5156   × cxp 5247  ^◡ccnv 5248   ↾ cres 5251   ∘ ccom 5253  Rel wrel 5254  ℝ^*cxr 10275   ≤ cle 10277  PosetRelcps 17406   TosetRel ctsr 17407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-pre-lttri 10212  ax-pre-lttrn 10213

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-ps 17408  df-tsr 17409

This theorem is referenced by:  cnfldle  19970  cnfldfun  19973  cnfldfunALT  19974  letopon  21230  leordtval2  21237  leordtval  21238  iccordt  21239  ordtrestixx  21247  xrge0tsms  22857  icopnfhmeo  22962  iccpnfhmeo  22964  xrhmeo  22965  xrhaus  29875  xrge0tsmsd  30125  cnvordtrestixx  30299  xrmulc1cn  30316  xrge0iifhmeo  30322  poimir  33775