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Mirrors > Home > MPE Home > Th. List > onelss | Structured version Visualization version GIF version |
Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
onelss | ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6374 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordelss 6380 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) | |
3 | 2 | ex 413 | . 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3948 Ord word 6363 Oncon0 6364 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-v 3476 df-in 3955 df-ss 3965 df-uni 4909 df-tr 5266 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 df-on 6368 |
This theorem is referenced by: ordunidif 6413 onelssi 6479 ssorduni 7768 sucexeloniOLD 7800 suceloniOLD 7802 tfisi 7850 poseq 8146 tfrlem9 8387 tfrlem11 8390 oaordex 8560 oaass 8563 odi 8581 omass 8582 oewordri 8594 nnaordex 8640 domtriord 9125 hartogs 9541 card2on 9551 tskwe 9947 infxpenlem 10010 cfub 10246 cfsuc 10254 coflim 10258 hsmexlem2 10424 ondomon 10560 pwcfsdom 10580 inar1 10772 tskord 10777 grudomon 10814 gruina 10815 sltres 27389 nosupno 27430 nosupbday 27432 noinfno 27445 oldssmade 27597 madebday 27619 mulsproplem13 27811 mulsproplem14 27812 dfrdg2 35059 aomclem6 42103 nnoeomeqom 42364 naddgeoa 42447 naddwordnexlem1 42450 naddwordnexlem4 42454 iscard5 42589 |
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