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Theorem xrge0infss 29995
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 3758 . . . . . . 7 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦𝐴) → 𝑦 ∈ (0[,]+∞))
2 0xr 10344 . . . . . . . . 9 0 ∈ ℝ*
3 pnfxr 10350 . . . . . . . . 9 +∞ ∈ ℝ*
4 iccgelb 12437 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
52, 3, 4mp3an12 1575 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6 eliccxr 12467 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
7 xrlenlt 10361 . . . . . . . . 9 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
82, 6, 7sylancr 581 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
95, 8mpbid 223 . . . . . . 7 (𝑦 ∈ (0[,]+∞) → ¬ 𝑦 < 0)
101, 9syl 17 . . . . . 6 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦𝐴) → ¬ 𝑦 < 0)
1110ralrimiva 3113 . . . . 5 (𝐴 ⊆ (0[,]+∞) → ∀𝑦𝐴 ¬ 𝑦 < 0)
1211ad3antrrr 721 . . . 4 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∀𝑦𝐴 ¬ 𝑦 < 0)
13 iccssxr 12463 . . . . . . . . . 10 (0[,]+∞) ⊆ ℝ*
14 ssralv 3828 . . . . . . . . . 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 791 . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . 14 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ (0[,]+∞))
1913, 18sseldi 3761 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ*)
20 simpllr 793 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 ≤ 0)
21 simpr 477 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 < 𝑦)
2216, 17, 19, 20, 21xrlelttrd 12198 . . . . . . . . . . . 12 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 < 𝑦)
2322ex 401 . . . . . . . . . . 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 3109 . . . . . . . . 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 705 . . . . . . 7 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2827imp 395 . . . . . 6 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
2928adantrl 707 . . . . 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 642 . . . 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 12492 . . . . 5 0 ∈ (0[,]+∞)
32 breq2 4815 . . . . . . . . 9 (𝑥 = 0 → (𝑦 < 𝑥𝑦 < 0))
3332notbid 309 . . . . . . . 8 (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
3433ralbidv 3133 . . . . . . 7 (𝑥 = 0 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 0))
35 breq1 4814 . . . . . . . . 9 (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
3635imbi1d 332 . . . . . . . 8 (𝑥 = 0 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3736ralbidv 3133 . . . . . . 7 (𝑥 = 0 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3834, 37anbi12d 624 . . . . . 6 (𝑥 = 0 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
3938rspcev 3462 . . . . 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 681 . . . 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 579 . . 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 793 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 𝑤 ∈ ℝ*)
43 simpr 477 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 0 ≤ 𝑤)
44 elxrge0 12490 . . . . 5 (𝑤 ∈ (0[,]+∞) ↔ (𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤))
4542, 43, 44sylanbrc 578 . . . 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 605 . . . . . . 7 (𝐴 ⊆ (0[,]+∞) → ((∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
4847adantr 472 . . . . . 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 395 . . . . 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 472 . . . 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 4815 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 < 𝑥𝑦 < 𝑤))
5251notbid 309 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
5352ralbidv 3133 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 𝑤))
54 breq1 4814 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 < 𝑦𝑤 < 𝑦))
5554imbi1d 332 . . . . . . 7 (𝑥 = 𝑤 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5655ralbidv 3133 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5753, 56anbi12d 624 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
5857rspcev 3462 . . . 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 579 . . 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 785 . . . 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 12191 . . . 4 ((𝑤 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6360, 61, 62syl2anc 579 . . 3 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6441, 59, 63mpjaodan 981 . 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 3771 . . . 4 ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
6613, 65mpan2 682 . . 3 (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67 xrinfmss 12347 . . 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 3225 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 197  wa 384  wo 873   = wceq 1652  wcel 2155  wral 3055  wrex 3056  wss 3734   class class class wbr 4811  (class class class)co 6846  0cc0 10193  +∞cpnf 10329  *cxr 10331   < clt 10332  cle 10333  [,]cicc 12385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-po 5200  df-so 5201  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-icc 12389
This theorem is referenced by:  xrge0infssd  29996  infxrge0lb  29999  infxrge0glb  30000  infxrge0gelb  30001  omsf  30826  omssubaddlem  30829  omssubadd  30830
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