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Theorem bnj1228 33960
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 33959 . 2 ((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
2 nfv 1918 . . . 4 𝑧(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
3 bnj1228.1 . . . . . . 7 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
43nfcii 2888 . . . . . 6 𝑥𝐵
54nfcri 2891 . . . . 5 𝑥 𝑧𝐵
6 nfv 1918 . . . . . 6 𝑥 ¬ 𝑦𝑅𝑧
74, 6nfralw 3309 . . . . 5 𝑥𝑦𝐵 ¬ 𝑦𝑅𝑧
85, 7nfan 1903 . . . 4 𝑥(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)
9 eleq1w 2817 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
10 breq2 5151 . . . . . . 7 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
1110notbid 318 . . . . . 6 (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
1211ralbidv 3178 . . . . 5 (𝑥 = 𝑧 → (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
139, 12anbi12d 632 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥) ↔ (𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)))
142, 8, 13cbvexv1 2339 . . 3 (∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥) ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
15 df-rex 3072 . . 3 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
16 df-rex 3072 . . 3 (∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
1714, 15, 163bitr4i 303 . 2 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
181, 17sylibr 233 1 ((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088  wal 1540  wex 1782  wcel 2107  wne 2941  wral 3062  wrex 3071  wss 3947  c0 4321   class class class wbr 5147   FrSe w-bnj15 33641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-reg 9583  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7851  df-1o 8461  df-bnj17 33636  df-bnj14 33638  df-bnj13 33640  df-bnj15 33642  df-bnj18 33644  df-bnj19 33646
This theorem is referenced by:  bnj1204  33961  bnj1311  33973  bnj1312  34007
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