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Mirrors > Home > MPE Home > Th. List > map0g | Structured version Visualization version GIF version |
Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
map0g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3890 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝐴) | |
2 | fconst6g 6007 | . . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}):𝐵⟶𝐴) | |
3 | elmapg 7757 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴 ↑𝑚 𝐵) ↔ (𝐵 × {𝑓}):𝐵⟶𝐴)) | |
4 | 2, 3 | syl5ibr 235 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}) ∈ (𝐴 ↑𝑚 𝐵))) |
5 | ne0i 3880 | . . . . . . 7 ⊢ ((𝐵 × {𝑓}) ∈ (𝐴 ↑𝑚 𝐵) → (𝐴 ↑𝑚 𝐵) ≠ ∅) | |
6 | 4, 5 | syl6 34 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐴 ↑𝑚 𝐵) ≠ ∅)) |
7 | 6 | exlimdv 1848 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓 ∈ 𝐴 → (𝐴 ↑𝑚 𝐵) ≠ ∅)) |
8 | 1, 7 | syl5bi 231 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ ∅ → (𝐴 ↑𝑚 𝐵) ≠ ∅)) |
9 | 8 | necon4d 2806 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → 𝐴 = ∅)) |
10 | f0 5999 | . . . . . . 7 ⊢ ∅:∅⟶𝐴 | |
11 | feq2 5940 | . . . . . . 7 ⊢ (𝐵 = ∅ → (∅:𝐵⟶𝐴 ↔ ∅:∅⟶𝐴)) | |
12 | 10, 11 | mpbiri 247 | . . . . . 6 ⊢ (𝐵 = ∅ → ∅:𝐵⟶𝐴) |
13 | elmapg 7757 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∅ ∈ (𝐴 ↑𝑚 𝐵) ↔ ∅:𝐵⟶𝐴)) | |
14 | 12, 13 | syl5ibr 235 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴 ↑𝑚 𝐵))) |
15 | ne0i 3880 | . . . . 5 ⊢ (∅ ∈ (𝐴 ↑𝑚 𝐵) → (𝐴 ↑𝑚 𝐵) ≠ ∅) | |
16 | 14, 15 | syl6 34 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → (𝐴 ↑𝑚 𝐵) ≠ ∅)) |
17 | 16 | necon2d 2805 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → 𝐵 ≠ ∅)) |
18 | 9, 17 | jcad 554 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅))) |
19 | oveq1 6556 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑚 𝐵) = (∅ ↑𝑚 𝐵)) | |
20 | map0b 7782 | . . 3 ⊢ (𝐵 ≠ ∅ → (∅ ↑𝑚 𝐵) = ∅) | |
21 | 19, 20 | sylan9eq 2664 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴 ↑𝑚 𝐵) = ∅) |
22 | 18, 21 | impbid1 214 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 {csn 4125 × cxp 5036 ⟶wf 5800 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 |
This theorem is referenced by: map0 7784 mapdom2 8016 |
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