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Mirrors > Home > MPE Home > Th. List > oe0m1 | Structured version Visualization version GIF version |
Description: Ordinal exponentiation with zero mantissa and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse. (Contributed by NM, 5-Jan-2005.) |
Ref | Expression |
---|---|
oe0m1 | ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑o 𝐴) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6169 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordgt0ge1 8105 | . . 3 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) |
4 | ssdif0 4277 | . . 3 ⊢ (1o ⊆ 𝐴 ↔ (1o ∖ 𝐴) = ∅) | |
5 | oe0m 8126 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴)) | |
6 | 5 | eqeq1d 2800 | . . 3 ⊢ (𝐴 ∈ On → ((∅ ↑o 𝐴) = ∅ ↔ (1o ∖ 𝐴) = ∅)) |
7 | 4, 6 | bitr4id 293 | . 2 ⊢ (𝐴 ∈ On → (1o ⊆ 𝐴 ↔ (∅ ↑o 𝐴) = ∅)) |
8 | 3, 7 | bitrd 282 | 1 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑o 𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 Ord word 6158 Oncon0 6159 (class class class)co 7135 1oc1o 8078 ↑o coe 8084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oexp 8091 |
This theorem is referenced by: oev2 8131 oesuclem 8133 oecl 8145 oewordri 8201 oelim2 8204 oeoa 8206 oeoe 8208 cantnf 9140 |
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