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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 7795 | . 2 ⊢ Ord On | |
2 | ordeleqon 7800 | . . . . 5 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | 2 | biimpi 216 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On)) |
4 | 3 | adantr 480 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On)) |
5 | ordsseleq 6414 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ⊆ On ↔ (𝐴 ∈ On ∨ 𝐴 = On))) | |
6 | 4, 5 | mpbird 257 | . 2 ⊢ ((Ord 𝐴 ∧ Ord On) → 𝐴 ⊆ On) |
7 | 1, 6 | mpan2 691 | 1 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 Ord word 6384 Oncon0 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 |
This theorem is referenced by: dford5 7802 onss 7803 orduni 7808 ordsuci 7827 ordsucuniel 7843 ordsucuni 7848 iordsmo 8395 dfrecs3 8410 dfrecs3OLD 8411 tfr2b 8434 tz7.44-2 8445 ordiso2 9552 ordtypelem7 9561 ordtypelem8 9562 oiid 9578 r1tr 9813 r1ordg 9815 r1ord3g 9816 r1pwss 9821 r1val1 9823 rankwflemb 9830 r1elwf 9833 rankr1ai 9835 cflim2 10300 cfss 10302 cfslb 10303 cfslbn 10304 cfslb2n 10305 cofsmo 10306 coftr 10310 inaprc 10873 nosepon 27724 satfn 35339 rdgprc 35775 limsucncmpi 36427 limexissup 43270 limexissupab 43272 nadd2rabord 43374 nadd1rabord 43378 |
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