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Mirrors > Home > MPE Home > Th. List > wess | Structured version Visualization version GIF version |
Description: Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31. (Contributed by NM, 19-Apr-1994.) |
Ref | Expression |
---|---|
wess | ⊢ (𝐴 ⊆ 𝐵 → (𝑅 We 𝐵 → 𝑅 We 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frss 5081 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Fr 𝐵 → 𝑅 Fr 𝐴)) | |
2 | soss 5053 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴)) | |
3 | 1, 2 | anim12d 586 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵) → (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))) |
4 | df-we 5075 | . 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵)) | |
5 | df-we 5075 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
6 | 3, 4, 5 | 3imtr4g 285 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 We 𝐵 → 𝑅 We 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ⊆ wss 3574 Or wor 5034 Fr wfr 5070 We wwe 5072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-ral 2917 df-in 3581 df-ss 3588 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 |
This theorem is referenced by: wefrc 5108 trssord 5740 ordelord 5745 omsinds 7084 fnwelem 7292 wfrlem5 7419 dfrecs3 7469 ordtypelem8 8430 oismo 8445 cantnfcl 8564 infxpenlem 8836 ac10ct 8857 dfac12lem2 8966 cflim2 9085 cofsmo 9091 hsmexlem1 9248 smobeth 9408 canthwelem 9472 gruina 9640 ltwefz 12762 dford5 31608 welb 33531 dnwech 37618 aomclem4 37627 dfac11 37632 onfrALTlem3 38759 onfrALTlem3VD 39123 |
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