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Mirrors > Home > MPE Home > Th. List > ordsson | Structured version Visualization version GIF version |
Description: Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
ordsson | ⊢ (Ord 𝐴 → 𝐴 ⊆ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordon 7498 | . 2 ⊢ Ord On | |
2 | ordeleqon 7503 | . . . . 5 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | 2 | biimpi 218 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On)) |
4 | 3 | adantr 483 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On)) |
5 | ordsseleq 6220 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ⊆ On ↔ (𝐴 ∈ On ∨ 𝐴 = On))) | |
6 | 4, 5 | mpbird 259 | . 2 ⊢ ((Ord 𝐴 ∧ Ord On) → 𝐴 ⊆ On) |
7 | 1, 6 | mpan2 689 | 1 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 Ord word 6190 Oncon0 6191 |
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-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 |
This theorem is referenced by: onss 7505 orduni 7509 ordsucuniel 7539 ordsucuni 7544 iordsmo 7994 dfrecs3 8009 tfr2b 8032 tz7.44-2 8043 ordiso2 8979 ordtypelem7 8988 ordtypelem8 8989 oiid 9005 r1tr 9205 r1ordg 9207 r1ord3g 9208 r1pwss 9213 r1val1 9215 rankwflemb 9222 r1elwf 9225 rankr1ai 9227 cflim2 9685 cfss 9687 cfslb 9688 cfslbn 9689 cfslb2n 9690 cofsmo 9691 coftr 9695 inaprc 10258 satfn 32602 dford5 32957 rdgprc 33039 nosepon 33172 limsucncmpi 33793 |
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