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Mirrors > Home > MPE Home > Th. List > tz6.26 | Structured version Visualization version GIF version |
Description: All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.) |
Ref | Expression |
---|---|
tz6.26 | ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wefr 5672 | . . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
2 | 1 | adantr 479 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴) |
3 | weso 5673 | . . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
4 | sopo 5613 | . . . . 5 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴) |
6 | 5 | adantr 479 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴) |
7 | simpr 483 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴) | |
8 | 2, 6, 7 | 3jca 1125 | . 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴)) |
9 | frpomin2 6352 | . 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) | |
10 | 8, 9 | sylan 578 | 1 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ≠ wne 2937 ∃wrex 3067 ⊆ wss 3949 ∅c0 4326 Po wpo 5592 Or wor 5593 Fr wfr 5634 Se wse 5635 We wwe 5636 Predcpred 6309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 |
This theorem is referenced by: tz6.26i 6360 wfiOLD 6362 wzel 35453 wsuclem 35454 |
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