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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 7797 | . 2 ⊢ Ord On | |
| 2 | ordeleqon 7802 | . . . . 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 6413 | . . 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 848 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: dford5 7804 onss 7805 orduni 7809 ordsuci 7828 ordsucuniel 7844 ordsucuni 7849 iordsmo 8397 dfrecs3 8412 dfrecs3OLD 8413 tfr2b 8436 tz7.44-2 8447 ordiso2 9555 ordtypelem7 9564 ordtypelem8 9565 oiid 9581 r1tr 9816 r1ordg 9818 r1ord3g 9819 r1pwss 9824 r1val1 9826 rankwflemb 9833 r1elwf 9836 rankr1ai 9838 cflim2 10303 cfss 10305 cfslb 10306 cfslbn 10307 cfslb2n 10308 cofsmo 10309 coftr 10313 inaprc 10876 nosepon 27710 satfn 35360 rdgprc 35795 limsucncmpi 36446 limexissup 43294 limexissupab 43296 nadd2rabord 43398 nadd1rabord 43402 |
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