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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 6331 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordgt0ge1 8443 | . . 3 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) |
4 | ssdif0 4327 | . . 3 ⊢ (1o ⊆ 𝐴 ↔ (1o ∖ 𝐴) = ∅) | |
5 | oe0m 8468 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴)) | |
6 | 5 | eqeq1d 2735 | . . 3 ⊢ (𝐴 ∈ On → ((∅ ↑o 𝐴) = ∅ ↔ (1o ∖ 𝐴) = ∅)) |
7 | 4, 6 | bitr4id 290 | . 2 ⊢ (𝐴 ∈ On → (1o ⊆ 𝐴 ↔ (∅ ↑o 𝐴) = ∅)) |
8 | 3, 7 | bitrd 279 | 1 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑o 𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∖ cdif 3911 ⊆ wss 3914 ∅c0 4286 Ord word 6320 Oncon0 6321 (class class class)co 7361 1oc1o 8409 ↑o coe 8415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-suc 6327 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oexp 8422 |
This theorem is referenced by: oev2 8473 oesuclem 8475 oecl 8487 oewordri 8543 oelim2 8546 oeoa 8548 oeoe 8550 cantnf 9637 oe0suclim 41659 |
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