Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7500 | . 2 ⊢ Ord On | |
2 | ordeleqon 7505 | . . . . 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 6222 | . . 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 3938 Ord word 6192 Oncon0 6193 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 |
This theorem is referenced by: onss 7507 orduni 7511 ordsucuniel 7541 ordsucuni 7546 iordsmo 7996 dfrecs3 8011 tfr2b 8034 tz7.44-2 8045 ordiso2 8981 ordtypelem7 8990 ordtypelem8 8991 oiid 9007 r1tr 9207 r1ordg 9209 r1ord3g 9210 r1pwss 9215 r1val1 9217 rankwflemb 9224 r1elwf 9227 rankr1ai 9229 cflim2 9687 cfss 9689 cfslb 9690 cfslbn 9691 cfslb2n 9692 cofsmo 9693 coftr 9697 inaprc 10260 satfn 32604 dford5 32959 rdgprc 33041 nosepon 33174 limsucncmpi 33795 |
Copyright terms: Public domain | W3C validator |