Metamath Proof Explorer


Theorem oen0

Description: Ordinal exponentiation with a nonzero base is nonzero. Proposition 8.32 of TakeutiZaring p. 67. (Contributed by NM, 4-Jan-2005)

Ref Expression
Assertion oen0 ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ∅ ∈ ( 𝐴o 𝐵 ) )

Proof

Step Hyp Ref Expression
1 oveq2 ( 𝑥 = ∅ → ( 𝐴o 𝑥 ) = ( 𝐴o ∅ ) )
2 1 eleq2d ( 𝑥 = ∅ → ( ∅ ∈ ( 𝐴o 𝑥 ) ↔ ∅ ∈ ( 𝐴o ∅ ) ) )
3 oveq2 ( 𝑥 = 𝑦 → ( 𝐴o 𝑥 ) = ( 𝐴o 𝑦 ) )
4 3 eleq2d ( 𝑥 = 𝑦 → ( ∅ ∈ ( 𝐴o 𝑥 ) ↔ ∅ ∈ ( 𝐴o 𝑦 ) ) )
5 oveq2 ( 𝑥 = suc 𝑦 → ( 𝐴o 𝑥 ) = ( 𝐴o suc 𝑦 ) )
6 5 eleq2d ( 𝑥 = suc 𝑦 → ( ∅ ∈ ( 𝐴o 𝑥 ) ↔ ∅ ∈ ( 𝐴o suc 𝑦 ) ) )
7 oveq2 ( 𝑥 = 𝐵 → ( 𝐴o 𝑥 ) = ( 𝐴o 𝐵 ) )
8 7 eleq2d ( 𝑥 = 𝐵 → ( ∅ ∈ ( 𝐴o 𝑥 ) ↔ ∅ ∈ ( 𝐴o 𝐵 ) ) )
9 0lt1o ∅ ∈ 1o
10 oe0 ( 𝐴 ∈ On → ( 𝐴o ∅ ) = 1o )
11 9 10 eleqtrrid ( 𝐴 ∈ On → ∅ ∈ ( 𝐴o ∅ ) )
12 11 adantr ( ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) → ∅ ∈ ( 𝐴o ∅ ) )
13 oecl ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴o 𝑦 ) ∈ On )
14 omordi ( ( ( 𝐴 ∈ On ∧ ( 𝐴o 𝑦 ) ∈ On ) ∧ ∅ ∈ ( 𝐴o 𝑦 ) ) → ( ∅ ∈ 𝐴 → ( ( 𝐴o 𝑦 ) ·o ∅ ) ∈ ( ( 𝐴o 𝑦 ) ·o 𝐴 ) ) )
15 om0 ( ( 𝐴o 𝑦 ) ∈ On → ( ( 𝐴o 𝑦 ) ·o ∅ ) = ∅ )
16 15 eleq1d ( ( 𝐴o 𝑦 ) ∈ On → ( ( ( 𝐴o 𝑦 ) ·o ∅ ) ∈ ( ( 𝐴o 𝑦 ) ·o 𝐴 ) ↔ ∅ ∈ ( ( 𝐴o 𝑦 ) ·o 𝐴 ) ) )
17 16 ad2antlr ( ( ( 𝐴 ∈ On ∧ ( 𝐴o 𝑦 ) ∈ On ) ∧ ∅ ∈ ( 𝐴o 𝑦 ) ) → ( ( ( 𝐴o 𝑦 ) ·o ∅ ) ∈ ( ( 𝐴o 𝑦 ) ·o 𝐴 ) ↔ ∅ ∈ ( ( 𝐴o 𝑦 ) ·o 𝐴 ) ) )
18 14 17 sylibd ( ( ( 𝐴 ∈ On ∧ ( 𝐴o 𝑦 ) ∈ On ) ∧ ∅ ∈ ( 𝐴o 𝑦 ) ) → ( ∅ ∈ 𝐴 → ∅ ∈ ( ( 𝐴o 𝑦 ) ·o 𝐴 ) ) )
19 13 18 syldanl ( ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) ∧ ∅ ∈ ( 𝐴o 𝑦 ) ) → ( ∅ ∈ 𝐴 → ∅ ∈ ( ( 𝐴o 𝑦 ) ·o 𝐴 ) ) )
20 oesuc ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴o suc 𝑦 ) = ( ( 𝐴o 𝑦 ) ·o 𝐴 ) )
21 20 eleq2d ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) → ( ∅ ∈ ( 𝐴o suc 𝑦 ) ↔ ∅ ∈ ( ( 𝐴o 𝑦 ) ·o 𝐴 ) ) )
22 21 adantr ( ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) ∧ ∅ ∈ ( 𝐴o 𝑦 ) ) → ( ∅ ∈ ( 𝐴o suc 𝑦 ) ↔ ∅ ∈ ( ( 𝐴o 𝑦 ) ·o 𝐴 ) ) )
23 19 22 sylibrd ( ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) ∧ ∅ ∈ ( 𝐴o 𝑦 ) ) → ( ∅ ∈ 𝐴 → ∅ ∈ ( 𝐴o suc 𝑦 ) ) )
24 23 exp31 ( 𝐴 ∈ On → ( 𝑦 ∈ On → ( ∅ ∈ ( 𝐴o 𝑦 ) → ( ∅ ∈ 𝐴 → ∅ ∈ ( 𝐴o suc 𝑦 ) ) ) ) )
25 24 com12 ( 𝑦 ∈ On → ( 𝐴 ∈ On → ( ∅ ∈ ( 𝐴o 𝑦 ) → ( ∅ ∈ 𝐴 → ∅ ∈ ( 𝐴o suc 𝑦 ) ) ) ) )
26 25 com34 ( 𝑦 ∈ On → ( 𝐴 ∈ On → ( ∅ ∈ 𝐴 → ( ∅ ∈ ( 𝐴o 𝑦 ) → ∅ ∈ ( 𝐴o suc 𝑦 ) ) ) ) )
27 26 impd ( 𝑦 ∈ On → ( ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) → ( ∅ ∈ ( 𝐴o 𝑦 ) → ∅ ∈ ( 𝐴o suc 𝑦 ) ) ) )
28 0ellim ( Lim 𝑥 → ∅ ∈ 𝑥 )
29 eqimss2 ( ( 𝐴o ∅ ) = 1o → 1o ⊆ ( 𝐴o ∅ ) )
30 10 29 syl ( 𝐴 ∈ On → 1o ⊆ ( 𝐴o ∅ ) )
31 oveq2 ( 𝑦 = ∅ → ( 𝐴o 𝑦 ) = ( 𝐴o ∅ ) )
32 31 sseq2d ( 𝑦 = ∅ → ( 1o ⊆ ( 𝐴o 𝑦 ) ↔ 1o ⊆ ( 𝐴o ∅ ) ) )
33 32 rspcev ( ( ∅ ∈ 𝑥 ∧ 1o ⊆ ( 𝐴o ∅ ) ) → ∃ 𝑦𝑥 1o ⊆ ( 𝐴o 𝑦 ) )
34 28 30 33 syl2an ( ( Lim 𝑥𝐴 ∈ On ) → ∃ 𝑦𝑥 1o ⊆ ( 𝐴o 𝑦 ) )
35 ssiun ( ∃ 𝑦𝑥 1o ⊆ ( 𝐴o 𝑦 ) → 1o 𝑦𝑥 ( 𝐴o 𝑦 ) )
36 34 35 syl ( ( Lim 𝑥𝐴 ∈ On ) → 1o 𝑦𝑥 ( 𝐴o 𝑦 ) )
37 36 adantrr ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) → 1o 𝑦𝑥 ( 𝐴o 𝑦 ) )
38 vex 𝑥 ∈ V
39 oelim ( ( ( 𝐴 ∈ On ∧ ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) ∧ ∅ ∈ 𝐴 ) → ( 𝐴o 𝑥 ) = 𝑦𝑥 ( 𝐴o 𝑦 ) )
40 38 39 mpanlr1 ( ( ( 𝐴 ∈ On ∧ Lim 𝑥 ) ∧ ∅ ∈ 𝐴 ) → ( 𝐴o 𝑥 ) = 𝑦𝑥 ( 𝐴o 𝑦 ) )
41 40 anasss ( ( 𝐴 ∈ On ∧ ( Lim 𝑥 ∧ ∅ ∈ 𝐴 ) ) → ( 𝐴o 𝑥 ) = 𝑦𝑥 ( 𝐴o 𝑦 ) )
42 41 an12s ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) → ( 𝐴o 𝑥 ) = 𝑦𝑥 ( 𝐴o 𝑦 ) )
43 37 42 sseqtrrd ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) → 1o ⊆ ( 𝐴o 𝑥 ) )
44 limelon ( ( 𝑥 ∈ V ∧ Lim 𝑥 ) → 𝑥 ∈ On )
45 38 44 mpan ( Lim 𝑥𝑥 ∈ On )
46 oecl ( ( 𝐴 ∈ On ∧ 𝑥 ∈ On ) → ( 𝐴o 𝑥 ) ∈ On )
47 46 ancoms ( ( 𝑥 ∈ On ∧ 𝐴 ∈ On ) → ( 𝐴o 𝑥 ) ∈ On )
48 45 47 sylan ( ( Lim 𝑥𝐴 ∈ On ) → ( 𝐴o 𝑥 ) ∈ On )
49 eloni ( ( 𝐴o 𝑥 ) ∈ On → Ord ( 𝐴o 𝑥 ) )
50 ordgt0ge1 ( Ord ( 𝐴o 𝑥 ) → ( ∅ ∈ ( 𝐴o 𝑥 ) ↔ 1o ⊆ ( 𝐴o 𝑥 ) ) )
51 48 49 50 3syl ( ( Lim 𝑥𝐴 ∈ On ) → ( ∅ ∈ ( 𝐴o 𝑥 ) ↔ 1o ⊆ ( 𝐴o 𝑥 ) ) )
52 51 adantrr ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) → ( ∅ ∈ ( 𝐴o 𝑥 ) ↔ 1o ⊆ ( 𝐴o 𝑥 ) ) )
53 43 52 mpbird ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) → ∅ ∈ ( 𝐴o 𝑥 ) )
54 53 ex ( Lim 𝑥 → ( ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) → ∅ ∈ ( 𝐴o 𝑥 ) ) )
55 54 a1dd ( Lim 𝑥 → ( ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) → ( ∀ 𝑦𝑥 ∅ ∈ ( 𝐴o 𝑦 ) → ∅ ∈ ( 𝐴o 𝑥 ) ) ) )
56 2 4 6 8 12 27 55 tfinds3 ( 𝐵 ∈ On → ( ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) → ∅ ∈ ( 𝐴o 𝐵 ) ) )
57 56 expd ( 𝐵 ∈ On → ( 𝐴 ∈ On → ( ∅ ∈ 𝐴 → ∅ ∈ ( 𝐴o 𝐵 ) ) ) )
58 57 com12 ( 𝐴 ∈ On → ( 𝐵 ∈ On → ( ∅ ∈ 𝐴 → ∅ ∈ ( 𝐴o 𝐵 ) ) ) )
59 58 imp31 ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ∅ ∈ ( 𝐴o 𝐵 ) )