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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 5515 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Fr 𝐵 → 𝑅 Fr 𝐴)) | |
2 | soss 5486 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴)) | |
3 | 1, 2 | anim12d 608 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵) → (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))) |
4 | df-we 5509 | . 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵)) | |
5 | df-we 5509 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
6 | 3, 4, 5 | 3imtr4g 297 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 We 𝐵 → 𝑅 We 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ⊆ wss 3933 Or wor 5466 Fr wfr 5504 We wwe 5506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-ral 3140 df-in 3940 df-ss 3949 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 |
This theorem is referenced by: wefrc 5542 trssord 6201 ordelord 6206 omsinds 7589 fnwelem 7814 wfrlem5 7948 dfrecs3 7998 ordtypelem8 8977 oismo 8992 cantnfcl 9118 infxpenlem 9427 ac10ct 9448 dfac12lem2 9558 cflim2 9673 cofsmo 9679 hsmexlem1 9836 smobeth 9996 canthwelem 10060 gruina 10228 ltwefz 13319 dford5 32854 welb 34892 dnwech 39526 aomclem4 39535 dfac11 39540 onfrALTlem3 40755 onfrALTlem3VD 41098 |
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