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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 5644 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Fr 𝐵 → 𝑅 Fr 𝐴)) | |
2 | soss 5609 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴)) | |
3 | 1, 2 | anim12d 610 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵) → (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))) |
4 | df-we 5634 | . 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵)) | |
5 | df-we 5634 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
6 | 3, 4, 5 | 3imtr4g 296 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 We 𝐵 → 𝑅 We 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ⊆ wss 3949 Or wor 5588 Fr wfr 5629 We wwe 5631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-v 3477 df-in 3956 df-ss 3966 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 |
This theorem is referenced by: wefrc 5671 trssord 6382 ordelord 6387 dford5 7771 omsinds 7876 omsindsOLD 7877 fnwelem 8117 wfrlem5OLD 8313 dfrecs3 8372 dfrecs3OLD 8373 ordtypelem8 9520 oismo 9535 cantnfcl 9662 infxpenlem 10008 ac10ct 10029 dfac12lem2 10139 cflim2 10258 cofsmo 10264 hsmexlem1 10421 smobeth 10581 canthwelem 10645 gruina 10813 ltwefz 13928 welb 36604 dnwech 41790 aomclem4 41799 dfac11 41804 oaun3lem1 42124 onfrALTlem3 43305 onfrALTlem3VD 43648 |
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