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Mirrors > Home > MPE Home > Th. List > ordgt0ge1 | Structured version Visualization version GIF version |
Description: Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.) |
Ref | Expression |
---|---|
ordgt0ge1 | ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6244 | . . 3 ⊢ ∅ ∈ On | |
2 | ordelsuc 7535 | . . 3 ⊢ ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) |
4 | df-1o 8102 | . . 3 ⊢ 1o = suc ∅ | |
5 | 4 | sseq1i 3995 | . 2 ⊢ (1o ⊆ 𝐴 ↔ suc ∅ ⊆ 𝐴) |
6 | 3, 5 | syl6bbr 291 | 1 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 Ord word 6190 Oncon0 6191 suc csuc 6193 1oc1o 8095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 df-suc 6197 df-1o 8102 |
This theorem is referenced by: ordge1n0 8123 oe0m1 8146 omword1 8199 omword2 8200 omlimcl 8204 oen0 8212 oewordi 8217 |
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