Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrge0infss Structured version   Visualization version   GIF version

Theorem xrge0infss 31447
Description: Any subset of nonnegative extended reals has an infimum. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
Assertion
Ref Expression
xrge0infss (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem xrge0infss
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ssel2 3938 . . . . . . 7 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦𝐴) → 𝑦 ∈ (0[,]+∞))
2 0xr 11136 . . . . . . . . 9 0 ∈ ℝ*
3 pnfxr 11143 . . . . . . . . 9 +∞ ∈ ℝ*
4 iccgelb 13249 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
52, 3, 4mp3an12 1452 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6 eliccxr 13281 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
7 xrlenlt 11154 . . . . . . . . 9 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
82, 6, 7sylancr 588 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
95, 8mpbid 231 . . . . . . 7 (𝑦 ∈ (0[,]+∞) → ¬ 𝑦 < 0)
101, 9syl 17 . . . . . 6 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦𝐴) → ¬ 𝑦 < 0)
1110ralrimiva 3142 . . . . 5 (𝐴 ⊆ (0[,]+∞) → ∀𝑦𝐴 ¬ 𝑦 < 0)
1211ad3antrrr 729 . . . 4 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∀𝑦𝐴 ¬ 𝑦 < 0)
13 iccssxr 13276 . . . . . . . . . 10 (0[,]+∞) ⊆ ℝ*
14 ssralv 4009 . . . . . . . . . 10 ((0[,]+∞) ⊆ ℝ* → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
1513, 14ax-mp 5 . . . . . . . . 9 (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
16 simplll 774 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 ∈ ℝ*)
172a1i 11 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 ∈ ℝ*)
18 simplr 768 . . . . . . . . . . . . . 14 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ (0[,]+∞))
1913, 18sselid 3941 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ*)
20 simpllr 775 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 ≤ 0)
21 simpr 486 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 < 𝑦)
2216, 17, 19, 20, 21xrlelttrd 13008 . . . . . . . . . . . 12 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 < 𝑦)
2322ex 414 . . . . . . . . . . 11 (((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) → (0 < 𝑦𝑤 < 𝑦))
2423imim1d 82 . . . . . . . . . 10 (((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) → ((𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → (0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2524ralimdva 3163 . . . . . . . . 9 ((𝑤 ∈ ℝ*𝑤 ≤ 0) → (∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2615, 25syl5 34 . . . . . . . 8 ((𝑤 ∈ ℝ*𝑤 ≤ 0) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2726adantll 713 . . . . . . 7 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2827imp 408 . . . . . 6 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
2928adantrl 715 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
3029an32s 651 . . . 4 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
31 0e0iccpnf 13305 . . . . 5 0 ∈ (0[,]+∞)
32 breq2 5108 . . . . . . . . 9 (𝑥 = 0 → (𝑦 < 𝑥𝑦 < 0))
3332notbid 318 . . . . . . . 8 (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
3433ralbidv 3173 . . . . . . 7 (𝑥 = 0 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 0))
35 breq1 5107 . . . . . . . . 9 (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
3635imbi1d 342 . . . . . . . 8 (𝑥 = 0 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3736ralbidv 3173 . . . . . . 7 (𝑥 = 0 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3834, 37anbi12d 632 . . . . . 6 (𝑥 = 0 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
3938rspcev 3580 . . . . 5 ((0 ∈ (0[,]+∞) ∧ (∀𝑦𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
4031, 39mpan 689 . . . 4 ((∀𝑦𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
4112, 30, 40syl2anc 585 . . 3 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
42 simpllr 775 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 𝑤 ∈ ℝ*)
43 simpr 486 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 0 ≤ 𝑤)
44 elxrge0 13303 . . . . 5 (𝑤 ∈ (0[,]+∞) ↔ (𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤))
4542, 43, 44sylanbrc 584 . . . 4 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 𝑤 ∈ (0[,]+∞))
4615a1i 11 . . . . . . . 8 (𝐴 ⊆ (0[,]+∞) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
4746anim2d 613 . . . . . . 7 (𝐴 ⊆ (0[,]+∞) → ((∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
4847adantr 482 . . . . . 6 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) → ((∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
4948imp 408 . . . . 5 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5049adantr 482 . . . 4 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
51 breq2 5108 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 < 𝑥𝑦 < 𝑤))
5251notbid 318 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
5352ralbidv 3173 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 𝑤))
54 breq1 5107 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 < 𝑦𝑤 < 𝑦))
5554imbi1d 342 . . . . . . 7 (𝑥 = 𝑤 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5655ralbidv 3173 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5753, 56anbi12d 632 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
5857rspcev 3580 . . . 4 ((𝑤 ∈ (0[,]+∞) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5945, 50, 58syl2anc 585 . . 3 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
60 simplr 768 . . . 4 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → 𝑤 ∈ ℝ*)
612a1i 11 . . . 4 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → 0 ∈ ℝ*)
62 xrletri 13001 . . . 4 ((𝑤 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6360, 61, 62syl2anc 585 . . 3 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6441, 59, 63mpjaodan 958 . 2 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
65 sstr 3951 . . . 4 ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
6613, 65mpan2 690 . . 3 (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67 xrinfmss 13158 . . 3 (𝐴 ⊆ ℝ* → ∃𝑤 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
6866, 67syl 17 . 2 (𝐴 ⊆ (0[,]+∞) → ∃𝑤 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
6964, 68r19.29a 3158 1 (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3063  wrex 3072  wss 3909   class class class wbr 5104  (class class class)co 7350  0cc0 10985  +∞cpnf 11120  *cxr 11122   < clt 11123  cle 11124  [,]cicc 13196
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 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-cnex 11041  ax-resscn 11042  ax-1cn 11043  ax-icn 11044  ax-addcl 11045  ax-addrcl 11046  ax-mulcl 11047  ax-mulrcl 11048  ax-mulcom 11049  ax-addass 11050  ax-mulass 11051  ax-distr 11052  ax-i2m1 11053  ax-1ne0 11054  ax-1rid 11055  ax-rnegex 11056  ax-rrecex 11057  ax-cnre 11058  ax-pre-lttri 11059  ax-pre-lttrn 11060  ax-pre-ltadd 11061  ax-pre-mulgt0 11062  ax-pre-sup 11063
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-po 5543  df-so 5544  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-1st 7912  df-2nd 7913  df-er 8582  df-en 8818  df-dom 8819  df-sdom 8820  df-pnf 11125  df-mnf 11126  df-xr 11127  df-ltxr 11128  df-le 11129  df-sub 11321  df-neg 11322  df-icc 13200
This theorem is referenced by:  xrge0infssd  31448  infxrge0lb  31451  infxrge0glb  31452  infxrge0gelb  31453  omsf  32657  omssubaddlem  32660  omssubadd  32661
  Copyright terms: Public domain W3C validator