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Mirrors > Home > MPE Home > Th. List > tz6.26i | Structured version Visualization version GIF version |
Description: All nonempty subclasses of a class having a well-founded set-like relation 𝑅 have 𝑅-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
tz6.26i.1 | ⊢ 𝑅 We 𝐴 |
tz6.26i.2 | ⊢ 𝑅 Se 𝐴 |
Ref | Expression |
---|---|
tz6.26i | ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tz6.26i.1 | . 2 ⊢ 𝑅 We 𝐴 | |
2 | tz6.26i.2 | . 2 ⊢ 𝑅 Se 𝐴 | |
3 | tz6.26 6174 | . 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) | |
4 | 1, 2, 3 | mpanl12 700 | 1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ≠ wne 3016 ∃wrex 3139 ⊆ wss 3936 ∅c0 4291 Se wse 5507 We wwe 5508 Predcpred 6142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 |
This theorem is referenced by: wfrlem16 7964 |
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