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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 11018 | . . . 4 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) | |
2 | n0 4348 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝐴) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝐴 ∈ P → ∃𝑓 𝑓 ∈ 𝐴) |
4 | prn0 11018 | . . . 4 ⊢ (𝐵 ∈ P → 𝐵 ≠ ∅) | |
5 | n0 4348 | . . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ 𝐵) | |
6 | 4, 5 | sylib 217 | . . 3 ⊢ (𝐵 ∈ P → ∃𝑔 𝑔 ∈ 𝐵) |
7 | 3, 6 | anim12i 611 | . 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 11030 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐵) → (𝑓𝐺𝑔) ∈ (𝐴𝐹𝐵))) |
11 | ne0i 4336 | . . . . . . . . 9 ⊢ ((𝑓𝐺𝑔) ∈ (𝐴𝐹𝐵) → (𝐴𝐹𝐵) ≠ ∅) | |
12 | 0pss 4446 | . . . . . . . . 9 ⊢ (∅ ⊊ (𝐴𝐹𝐵) ↔ (𝐴𝐹𝐵) ≠ ∅) | |
13 | 11, 12 | sylibr 233 | . . . . . . . 8 ⊢ ((𝑓𝐺𝑔) ∈ (𝐴𝐹𝐵) → ∅ ⊊ (𝐴𝐹𝐵)) |
14 | 10, 13 | syl6 35 | . . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐵) → ∅ ⊊ (𝐴𝐹𝐵))) |
15 | 14 | expcomd 415 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑔 ∈ 𝐵 → (𝑓 ∈ 𝐴 → ∅ ⊊ (𝐴𝐹𝐵)))) |
16 | 15 | exlimdv 1928 | . . . . 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 1928 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑓 𝑓 ∈ 𝐴 → (∃𝑔 𝑔 ∈ 𝐵 → ∅ ⊊ (𝐴𝐹𝐵)))) |
19 | 18 | impd 409 | . 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 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2704 ≠ wne 2936 ∃wrex 3066 ⊊ wpss 3948 ∅c0 4324 (class class class)co 7424 ∈ cmpo 7426 Qcnq 10881 Pcnp 10888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-ni 10901 df-nq 10941 df-np 11010 |
This theorem is referenced by: genpcl 11037 |
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