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Theorem xrge0infss 29865
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 3747 . . . . . . 7 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦𝐴) → 𝑦 ∈ (0[,]+∞))
2 0xr 10288 . . . . . . . . 9 0 ∈ ℝ*
3 pnfxr 10294 . . . . . . . . 9 +∞ ∈ ℝ*
4 iccgelb 12435 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
52, 3, 4mp3an12 1562 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6 eliccxr 12465 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
7 xrlenlt 10305 . . . . . . . . 9 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
82, 6, 7sylancr 575 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
95, 8mpbid 222 . . . . . . 7 (𝑦 ∈ (0[,]+∞) → ¬ 𝑦 < 0)
101, 9syl 17 . . . . . 6 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦𝐴) → ¬ 𝑦 < 0)
1110ralrimiva 3115 . . . . 5 (𝐴 ⊆ (0[,]+∞) → ∀𝑦𝐴 ¬ 𝑦 < 0)
1211ad3antrrr 709 . . . 4 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∀𝑦𝐴 ¬ 𝑦 < 0)
13 iccssxr 12461 . . . . . . . . . 10 (0[,]+∞) ⊆ ℝ*
14 ssralv 3815 . . . . . . . . . 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 758 . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . 14 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ (0[,]+∞))
1913, 18sseldi 3750 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ*)
20 simpllr 760 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 ≤ 0)
21 simpr 471 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 < 𝑦)
2216, 17, 19, 20, 21xrlelttrd 12196 . . . . . . . . . . . 12 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 < 𝑦)
2322ex 397 . . . . . . . . . . 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 3111 . . . . . . . . 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 693 . . . . . . 7 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2827imp 393 . . . . . 6 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
2928adantrl 695 . . . . 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 631 . . . 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 12490 . . . . 5 0 ∈ (0[,]+∞)
32 breq2 4790 . . . . . . . . 9 (𝑥 = 0 → (𝑦 < 𝑥𝑦 < 0))
3332notbid 307 . . . . . . . 8 (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
3433ralbidv 3135 . . . . . . 7 (𝑥 = 0 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 0))
35 breq1 4789 . . . . . . . . 9 (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
3635imbi1d 330 . . . . . . . 8 (𝑥 = 0 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3736ralbidv 3135 . . . . . . 7 (𝑥 = 0 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3834, 37anbi12d 616 . . . . . 6 (𝑥 = 0 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
3938rspcev 3460 . . . . 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 670 . . . 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 573 . . 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 760 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 𝑤 ∈ ℝ*)
43 simpr 471 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 0 ≤ 𝑤)
44 elxrge0 12488 . . . . 5 (𝑤 ∈ (0[,]+∞) ↔ (𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤))
4542, 43, 44sylanbrc 572 . . . 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 599 . . . . . . 7 (𝐴 ⊆ (0[,]+∞) → ((∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
4847adantr 466 . . . . . 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 393 . . . . 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 466 . . . 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 4790 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 < 𝑥𝑦 < 𝑤))
5251notbid 307 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
5352ralbidv 3135 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 𝑤))
54 breq1 4789 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 < 𝑦𝑤 < 𝑦))
5554imbi1d 330 . . . . . . 7 (𝑥 = 𝑤 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5655ralbidv 3135 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5753, 56anbi12d 616 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
5857rspcev 3460 . . . 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 573 . . 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 752 . . . 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 12189 . . . 4 ((𝑤 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6360, 61, 62syl2anc 573 . . 3 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6441, 59, 63mpjaodan 943 . 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 3760 . . . 4 ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
6613, 65mpan2 671 . . 3 (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67 xrinfmss 12345 . . 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 3226 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 196  wa 382  wo 836   = wceq 1631  wcel 2145  wral 3061  wrex 3062  wss 3723   class class class wbr 4786  (class class class)co 6793  0cc0 10138  +∞cpnf 10273  *cxr 10275   < clt 10276  cle 10277  [,]cicc 12383
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-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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-reu 3068  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-iun 4656  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-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  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-sub 10470  df-neg 10471  df-icc 12387
This theorem is referenced by:  xrge0infssd  29866  infxrge0lb  29869  infxrge0glb  29870  infxrge0gelb  29871  omsf  30698  omssubaddlem  30701  omssubadd  30702
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