Theorem map0g 6857

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.)

Assertion

Ref	Expression
map0g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))

Proof of Theorem map0g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconst6g 5535	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}):𝐵⟶𝐴)
2		elmapg 6830	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴 ↑𝑚 𝐵) ↔ (𝐵 × {𝑓}):𝐵⟶𝐴))
3	1, 2	imbitrrid 156	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}) ∈ (𝐴 ↑𝑚 𝐵)))
4		ne0i 3501	. . . . . . 7 ⊢ ((𝐵 × {𝑓}) ∈ (𝐴 ↑𝑚 𝐵) → (𝐴 ↑𝑚 𝐵) ≠ ∅)
5	3, 4	syl6 33	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐴 ↑𝑚 𝐵) ≠ ∅))
6	5	exlimdv 1867	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓 ∈ 𝐴 → (𝐴 ↑𝑚 𝐵) ≠ ∅))
7	6	necon2bd 2460	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → ¬ ∃𝑓 𝑓 ∈ 𝐴))
8		notm0 3515	. . . 4 ⊢ (¬ ∃𝑓 𝑓 ∈ 𝐴 ↔ 𝐴 = ∅)
9	7, 8	imbitrdi 161	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → 𝐴 = ∅))
10		f0 5527	. . . . . . 7 ⊢ ∅:∅⟶𝐴
11		feq2 5466	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅:𝐵⟶𝐴 ↔ ∅:∅⟶𝐴))
12	10, 11	mpbiri 168	. . . . . 6 ⊢ (𝐵 = ∅ → ∅:𝐵⟶𝐴)
13		elmapg 6830	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∅ ∈ (𝐴 ↑𝑚 𝐵) ↔ ∅:𝐵⟶𝐴))
14	12, 13	imbitrrid 156	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴 ↑𝑚 𝐵)))
15		ne0i 3501	. . . . 5 ⊢ (∅ ∈ (𝐴 ↑𝑚 𝐵) → (𝐴 ↑𝑚 𝐵) ≠ ∅)
16	14, 15	syl6 33	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → (𝐴 ↑𝑚 𝐵) ≠ ∅))
17	16	necon2d 2461	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → 𝐵 ≠ ∅))
18	9, 17	jcad 307	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19		oveq1 6025	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑚 𝐵) = (∅ ↑𝑚 𝐵))
20		map0b 6856	. . 3 ⊢ (𝐵 ≠ ∅ → (∅ ↑𝑚 𝐵) = ∅)
21	19, 20	sylan9eq 2284	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴 ↑𝑚 𝐵) = ∅)
22	18, 21	impbid1 142	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202   ≠ wne 2402  ∅c0 3494  {csn 3669   × cxp 4723  ⟶wf 5322  (class class class)co 6018   ↑_𝑚 cmap 6817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-map 6819

This theorem is referenced by:  map0  6858