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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj69 | Structured version Visualization version GIF version |
Description: Existence of a minimal element in certain classes: if 𝑅 is well-founded and set-like on 𝐴, then every nonempty subclass of 𝐴 has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj69 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biid 253 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) ↔ (𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) | |
2 | biid 253 | . 2 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑥)) | |
3 | biid 253 | . 2 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | |
4 | 1, 2, 3 | bnj1189 31676 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1071 ∈ wcel 2106 ≠ wne 2968 ∀wral 3089 ∃wrex 3090 ⊆ wss 3791 ∅c0 4140 class class class wbr 4886 FrSe w-bnj15 31360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-reg 8786 ax-inf2 8835 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-om 7344 df-1o 7843 df-bnj17 31355 df-bnj14 31357 df-bnj13 31359 df-bnj15 31361 df-bnj18 31363 df-bnj19 31365 |
This theorem is referenced by: bnj1228 31678 bnj1523 31738 |
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