ordsson - Metamath Proof Explorer

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Theorem ordsson 7484

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.)

**Assertion**
Ref	Expression
ordsson	⊢ (Ord 𝐴 → 𝐴 ⊆ On)

**Proof of Theorem ordsson**
Step	Hyp	Ref	Expression
1		ordon 7478	. 2 ⊢ Ord On
2		ordeleqon 7483	. . . . 5 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3	2	biimpi 219	. . . 4 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
4	3	adantr 484	. . 3 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On))
5		ordsseleq 6188	. . 3 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ⊆ On ↔ (𝐴 ∈ On ∨ 𝐴 = On)))
6	4, 5	mpbird 260	. 2 ⊢ ((Ord 𝐴 ∧ Ord On) → 𝐴 ⊆ On)
7	1, 6	mpan2 690	1 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 Ord word 6158 Oncon0 6159

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163

This theorem is referenced by: onss 7485 orduni 7489 ordsucuniel 7519 ordsucuni 7524 iordsmo 7977 dfrecs3 7992 tfr2b 8015 tz7.44-2 8026 ordiso2 8963 ordtypelem7 8972 ordtypelem8 8973 oiid 8989 r1tr 9189 r1ordg 9191 r1ord3g 9192 r1pwss 9197 r1val1 9199 rankwflemb 9206 r1elwf 9209 rankr1ai 9211 cflim2 9674 cfss 9676 cfslb 9677 cfslbn 9678 cfslb2n 9679 cofsmo 9680 coftr 9684 inaprc 10247 satfn 32715 dford5 33070 rdgprc 33152 nosepon 33285 limsucncmpi 33906

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