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Mirrors > Home > MPE Home > Th. List > genpn0 | Structured version Visualization version GIF version |
Description: The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
genp.1 | ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)}) |
genp.2 | ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) |
Ref | Expression |
---|---|
genpn0 | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∅ ⊊ (𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prn0 10411 | . . . 4 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) | |
2 | n0 4310 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝐴) | |
3 | 1, 2 | sylib 220 | . . 3 ⊢ (𝐴 ∈ P → ∃𝑓 𝑓 ∈ 𝐴) |
4 | prn0 10411 | . . . 4 ⊢ (𝐵 ∈ P → 𝐵 ≠ ∅) | |
5 | n0 4310 | . . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ 𝐵) | |
6 | 4, 5 | sylib 220 | . . 3 ⊢ (𝐵 ∈ P → ∃𝑔 𝑔 ∈ 𝐵) |
7 | 3, 6 | anim12i 614 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑓 𝑓 ∈ 𝐴 ∧ ∃𝑔 𝑔 ∈ 𝐵)) |
8 | genp.1 | . . . . . . . . 9 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)}) | |
9 | genp.2 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) | |
10 | 8, 9 | genpprecl 10423 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐵) → (𝑓𝐺𝑔) ∈ (𝐴𝐹𝐵))) |
11 | ne0i 4300 | . . . . . . . . 9 ⊢ ((𝑓𝐺𝑔) ∈ (𝐴𝐹𝐵) → (𝐴𝐹𝐵) ≠ ∅) | |
12 | 0pss 4396 | . . . . . . . . 9 ⊢ (∅ ⊊ (𝐴𝐹𝐵) ↔ (𝐴𝐹𝐵) ≠ ∅) | |
13 | 11, 12 | sylibr 236 | . . . . . . . 8 ⊢ ((𝑓𝐺𝑔) ∈ (𝐴𝐹𝐵) → ∅ ⊊ (𝐴𝐹𝐵)) |
14 | 10, 13 | syl6 35 | . . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐵) → ∅ ⊊ (𝐴𝐹𝐵))) |
15 | 14 | expcomd 419 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑔 ∈ 𝐵 → (𝑓 ∈ 𝐴 → ∅ ⊊ (𝐴𝐹𝐵)))) |
16 | 15 | exlimdv 1934 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 𝑔 ∈ 𝐵 → (𝑓 ∈ 𝐴 → ∅ ⊊ (𝐴𝐹𝐵)))) |
17 | 16 | com23 86 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ 𝐴 → (∃𝑔 𝑔 ∈ 𝐵 → ∅ ⊊ (𝐴𝐹𝐵)))) |
18 | 17 | exlimdv 1934 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑓 𝑓 ∈ 𝐴 → (∃𝑔 𝑔 ∈ 𝐵 → ∅ ⊊ (𝐴𝐹𝐵)))) |
19 | 18 | impd 413 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((∃𝑓 𝑓 ∈ 𝐴 ∧ ∃𝑔 𝑔 ∈ 𝐵) → ∅ ⊊ (𝐴𝐹𝐵))) |
20 | 7, 19 | mpd 15 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∅ ⊊ (𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ≠ wne 3016 ∃wrex 3139 ⊊ wpss 3937 ∅c0 4291 (class class class)co 7156 ∈ cmpo 7158 Qcnq 10274 Pcnp 10281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-ni 10294 df-nq 10334 df-np 10403 |
This theorem is referenced by: genpcl 10430 |
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