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Theorem xrge0infss 30410
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 3959 . . . . . . 7 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦𝐴) → 𝑦 ∈ (0[,]+∞))
2 0xr 10676 . . . . . . . . 9 0 ∈ ℝ*
3 pnfxr 10683 . . . . . . . . 9 +∞ ∈ ℝ*
4 iccgelb 12781 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
52, 3, 4mp3an12 1442 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6 eliccxr 12811 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
7 xrlenlt 10694 . . . . . . . . 9 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
82, 6, 7sylancr 587 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
95, 8mpbid 233 . . . . . . 7 (𝑦 ∈ (0[,]+∞) → ¬ 𝑦 < 0)
101, 9syl 17 . . . . . 6 ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦𝐴) → ¬ 𝑦 < 0)
1110ralrimiva 3179 . . . . 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 12807 . . . . . . . . . 10 (0[,]+∞) ⊆ ℝ*
14 ssralv 4030 . . . . . . . . . 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, 18sseldi 3962 . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 13 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 < 𝑦)
2216, 17, 19, 20, 21xrlelttrd 12541 . . . . . . . . . . . 12 ((((𝑤 ∈ ℝ*𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 < 𝑦)
2322ex 413 . . . . . . . . . . 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 3174 . . . . . . . . 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 407 . . . . . 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 12835 . . . . 5 0 ∈ (0[,]+∞)
32 breq2 5061 . . . . . . . . 9 (𝑥 = 0 → (𝑦 < 𝑥𝑦 < 0))
3332notbid 319 . . . . . . . 8 (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
3433ralbidv 3194 . . . . . . 7 (𝑥 = 0 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 0))
35 breq1 5060 . . . . . . . . 9 (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
3635imbi1d 343 . . . . . . . 8 (𝑥 = 0 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3736ralbidv 3194 . . . . . . 7 (𝑥 = 0 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
3834, 37anbi12d 630 . . . . . 6 (𝑥 = 0 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
3938rspcev 3620 . . . . 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 584 . . 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 485 . . . . 5 ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 0 ≤ 𝑤)
44 elxrge0 12833 . . . . 5 (𝑤 ∈ (0[,]+∞) ↔ (𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤))
4542, 43, 44sylanbrc 583 . . . 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 611 . . . . . . 7 (𝐴 ⊆ (0[,]+∞) → ((∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
4847adantr 481 . . . . . 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 407 . . . . 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 481 . . . 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 5061 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 < 𝑥𝑦 < 𝑤))
5251notbid 319 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
5352ralbidv 3194 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦 < 𝑤))
54 breq1 5060 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 < 𝑦𝑤 < 𝑦))
5554imbi1d 343 . . . . . . 7 (𝑥 = 𝑤 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5655ralbidv 3194 . . . . . 6 (𝑥 = 𝑤 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
5753, 56anbi12d 630 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
5857rspcev 3620 . . . 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 584 . . 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 12534 . . . 4 ((𝑤 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6360, 61, 62syl2anc 584 . . 3 (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
6441, 59, 63mpjaodan 952 . 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 3972 . . . 4 ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
6613, 65mpan2 687 . . 3 (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67 xrinfmss 12691 . . 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 3286 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 207  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933   class class class wbr 5057  (class class class)co 7145  0cc0 10525  +∞cpnf 10660  *cxr 10662   < clt 10663  cle 10664  [,]cicc 12729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-icc 12733
This theorem is referenced by:  xrge0infssd  30411  infxrge0lb  30414  infxrge0glb  30415  infxrge0gelb  30416  omsf  31453  omssubaddlem  31456  omssubadd  31457
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