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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 5650 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordgt0ge1 7464 | . . 3 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜 ⊆ 𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1𝑜 ⊆ 𝐴)) |
4 | oe0m 7485 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ↑𝑜 𝐴) = (1𝑜 ∖ 𝐴)) | |
5 | 4 | eqeq1d 2612 | . . 3 ⊢ (𝐴 ∈ On → ((∅ ↑𝑜 𝐴) = ∅ ↔ (1𝑜 ∖ 𝐴) = ∅)) |
6 | ssdif0 3896 | . . 3 ⊢ (1𝑜 ⊆ 𝐴 ↔ (1𝑜 ∖ 𝐴) = ∅) | |
7 | 5, 6 | syl6rbbr 278 | . 2 ⊢ (𝐴 ∈ On → (1𝑜 ⊆ 𝐴 ↔ (∅ ↑𝑜 𝐴) = ∅)) |
8 | 3, 7 | bitrd 267 | 1 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑𝑜 𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 Ord word 5639 Oncon0 5640 (class class class)co 6549 1𝑜c1o 7440 ↑𝑜 coe 7446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oexp 7453 |
This theorem is referenced by: oev2 7490 oesuclem 7492 oecl 7504 oewordri 7559 oelim2 7562 oeoa 7564 oeoe 7566 cantnf 8473 |
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