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Theorem bnj1228 32393
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.)
Hypothesis
Ref Expression
bnj1228.1 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
Assertion
Ref Expression
bnj1228 ((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Distinct variable groups:   𝑦,𝐴   𝑤,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑤
Allowed substitution hints:   𝐴(𝑥,𝑤)   𝐵(𝑥)   𝑅(𝑤)

Proof of Theorem bnj1228
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj69 32392 . 2 ((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
2 nfv 1915 . . . 4 𝑧(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
3 bnj1228.1 . . . . . . 7 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
43nfcii 2940 . . . . . 6 𝑥𝐵
54nfcri 2943 . . . . 5 𝑥 𝑧𝐵
6 nfv 1915 . . . . . 6 𝑥 ¬ 𝑦𝑅𝑧
74, 6nfralw 3189 . . . . 5 𝑥𝑦𝐵 ¬ 𝑦𝑅𝑧
85, 7nfan 1900 . . . 4 𝑥(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)
9 eleq1w 2872 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
10 breq2 5034 . . . . . . 7 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
1110notbid 321 . . . . . 6 (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
1211ralbidv 3162 . . . . 5 (𝑥 = 𝑧 → (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
139, 12anbi12d 633 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥) ↔ (𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)))
142, 8, 13cbvexv1 2351 . . 3 (∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥) ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
15 df-rex 3112 . . 3 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
16 df-rex 3112 . . 3 (∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
1714, 15, 163bitr4i 306 . 2 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
181, 17sylibr 237 1 ((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084  wal 1536  wex 1781  wcel 2111  wne 2987  wral 3106  wrex 3107  wss 3881  c0 4243   class class class wbr 5030   FrSe w-bnj15 32072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-reg 9040  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-bnj17 32067  df-bnj14 32069  df-bnj13 32071  df-bnj15 32073  df-bnj18 32075  df-bnj19 32077
This theorem is referenced by:  bnj1204  32394  bnj1311  32406  bnj1312  32440
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