![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > oe0m1 | Structured version Visualization version GIF version |
Description: Ordinal exponentiation with zero base and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse. Definition 2.6 of [Schloeder] p. 4. (Contributed by NM, 5-Jan-2005.) |
Ref | Expression |
---|---|
oe0m1 | ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑o 𝐴) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6375 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordgt0ge1 8497 | . . 3 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) |
4 | ssdif0 4364 | . . 3 ⊢ (1o ⊆ 𝐴 ↔ (1o ∖ 𝐴) = ∅) | |
5 | oe0m 8522 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴)) | |
6 | 5 | eqeq1d 2732 | . . 3 ⊢ (𝐴 ∈ On → ((∅ ↑o 𝐴) = ∅ ↔ (1o ∖ 𝐴) = ∅)) |
7 | 4, 6 | bitr4id 289 | . 2 ⊢ (𝐴 ∈ On → (1o ⊆ 𝐴 ↔ (∅ ↑o 𝐴) = ∅)) |
8 | 3, 7 | bitrd 278 | 1 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑o 𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 Ord word 6364 Oncon0 6365 (class class class)co 7413 1oc1o 8463 ↑o coe 8469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-suc 6371 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7416 df-oprab 7417 df-mpo 7418 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oexp 8476 |
This theorem is referenced by: oev2 8527 oesuclem 8529 oecl 8541 oewordri 8596 oelim2 8599 oeoa 8601 oeoe 8603 cantnf 9692 oe0suclim 42331 |
Copyright terms: Public domain | W3C validator |