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Theorem xrge0infss 32240
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 3976 . . . . . . 7 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦𝐴) → 𝑦 ∈ (0[,]+∞))
2 0xr 11265 . . . . . . . . 9 0 ∈ ℝ*
3 pnfxr 11272 . . . . . . . . 9 +∞ ∈ ℝ*
4 iccgelb 13384 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
52, 3, 4mp3an12 1449 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6 eliccxr 13416 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
7 xrlenlt 11283 . . . . . . . . 9 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
82, 6, 7sylancr 585 . . . . . . . 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 3144 . . . . 5 (𝐴 ⊆ (0[,]+∞) → ∀𝑦𝐴 ¬ 𝑦 < 0)
1211ad3antrrr 726 . . . 4 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∀𝑦𝐴 ¬ 𝑦 < 0)
13 iccssxr 13411 . . . . . . . . . 10 (0[,]+∞) ⊆ ℝ*
14 ssralv 4049 . . . . . . . . . 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 771 . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . 14 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ (0[,]+∞))
1913, 18sselid 3979 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ*)
20 simpllr 772 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 ≤ 0)
21 simpr 483 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 < 𝑦)
2216, 17, 19, 20, 21xrlelttrd 13143 . . . . . . . . . . . 12 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 < 𝑦)
2322ex 411 . . . . . . . . . . 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 3165 . . . . . . . . 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 710 . . . . . . 7 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2827imp 405 . . . . . 6 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
2928adantrl 712 . . . . 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 648 . . . 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 13440 . . . . 5 0 ∈ (0[,]+∞)
32 breq2 5151 . . . . . . . . 9 (𝑥 = 0 → (𝑦 < 𝑥𝑦 < 0))
3332notbid 317 . . . . . . . 8 (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
3433ralbidv 3175 . . . . . . 7 (𝑥 = 0 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 0))
35 breq1 5150 . . . . . . . . 9 (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
3635imbi1d 340 . . . . . . . 8 (𝑥 = 0 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3736ralbidv 3175 . . . . . . 7 (𝑥 = 0 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3834, 37anbi12d 629 . . . . . 6 (𝑥 = 0 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
3938rspcev 3611 . . . . 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 686 . . . 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 582 . . 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 772 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 𝑤 ∈ ℝ*)
43 simpr 483 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 0 ≤ 𝑤)
44 elxrge0 13438 . . . . 5 (𝑤 ∈ (0[,]+∞) ↔ (𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤))
4542, 43, 44sylanbrc 581 . . . 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 610 . . . . . . 7 (𝐴 ⊆ (0[,]+∞) → ((∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
4847adantr 479 . . . . . 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 405 . . . . 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 479 . . . 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 5151 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 < 𝑥𝑦 < 𝑤))
5251notbid 317 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
5352ralbidv 3175 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 𝑤))
54 breq1 5150 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 < 𝑦𝑤 < 𝑦))
5554imbi1d 340 . . . . . . 7 (𝑥 = 𝑤 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5655ralbidv 3175 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5753, 56anbi12d 629 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
5857rspcev 3611 . . . 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 582 . . 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 765 . . . 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 13136 . . . 4 ((𝑤 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6360, 61, 62syl2anc 582 . . 3 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6441, 59, 63mpjaodan 955 . 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 3989 . . . 4 ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
6613, 65mpan2 687 . . 3 (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67 xrinfmss 13293 . . 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 3160 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 394  wo 843   = wceq 1539  wcel 2104  wral 3059  wrex 3068  wss 3947   class class class wbr 5147  (class class class)co 7411  0cc0 11112  +∞cpnf 11249  *cxr 11251   < clt 11252  cle 11253  [,]cicc 13331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-icc 13335
This theorem is referenced by:  xrge0infssd  32241  infxrge0lb  32244  infxrge0glb  32245  infxrge0gelb  32246  omsf  33593  omssubaddlem  33596  omssubadd  33597
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