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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 3931 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝐴) | |
2 | fconst6g 6094 | . . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}):𝐵⟶𝐴) | |
3 | elmapg 7870 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴 ↑𝑚 𝐵) ↔ (𝐵 × {𝑓}):𝐵⟶𝐴)) | |
4 | 2, 3 | syl5ibr 236 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}) ∈ (𝐴 ↑𝑚 𝐵))) |
5 | ne0i 3921 | . . . . . . 7 ⊢ ((𝐵 × {𝑓}) ∈ (𝐴 ↑𝑚 𝐵) → (𝐴 ↑𝑚 𝐵) ≠ ∅) | |
6 | 4, 5 | syl6 35 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐴 ↑𝑚 𝐵) ≠ ∅)) |
7 | 6 | exlimdv 1861 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓 ∈ 𝐴 → (𝐴 ↑𝑚 𝐵) ≠ ∅)) |
8 | 1, 7 | syl5bi 232 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ ∅ → (𝐴 ↑𝑚 𝐵) ≠ ∅)) |
9 | 8 | necon4d 2818 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → 𝐴 = ∅)) |
10 | f0 6086 | . . . . . . 7 ⊢ ∅:∅⟶𝐴 | |
11 | feq2 6027 | . . . . . . 7 ⊢ (𝐵 = ∅ → (∅:𝐵⟶𝐴 ↔ ∅:∅⟶𝐴)) | |
12 | 10, 11 | mpbiri 248 | . . . . . 6 ⊢ (𝐵 = ∅ → ∅:𝐵⟶𝐴) |
13 | elmapg 7870 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∅ ∈ (𝐴 ↑𝑚 𝐵) ↔ ∅:𝐵⟶𝐴)) | |
14 | 12, 13 | syl5ibr 236 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴 ↑𝑚 𝐵))) |
15 | ne0i 3921 | . . . . 5 ⊢ (∅ ∈ (𝐴 ↑𝑚 𝐵) → (𝐴 ↑𝑚 𝐵) ≠ ∅) | |
16 | 14, 15 | syl6 35 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → (𝐴 ↑𝑚 𝐵) ≠ ∅)) |
17 | 16 | necon2d 2817 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → 𝐵 ≠ ∅)) |
18 | 9, 17 | jcad 555 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅))) |
19 | oveq1 6657 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑚 𝐵) = (∅ ↑𝑚 𝐵)) | |
20 | map0b 7896 | . . 3 ⊢ (𝐵 ≠ ∅ → (∅ ↑𝑚 𝐵) = ∅) | |
21 | 19, 20 | sylan9eq 2676 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴 ↑𝑚 𝐵) = ∅) |
22 | 18, 21 | impbid1 215 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 {csn 4177 × cxp 5112 ⟶wf 5884 (class class class)co 6650 ↑𝑚 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 |
This theorem is referenced by: map0 7898 mapdom2 8131 |
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