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Theorem bnj1228 34550
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 34549 . 2 ((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
2 nfv 1909 . . . 4 𝑧(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
3 bnj1228.1 . . . . . . 7 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
43nfcii 2881 . . . . . 6 𝑥𝐵
54nfcri 2884 . . . . 5 𝑥 𝑧𝐵
6 nfv 1909 . . . . . 6 𝑥 ¬ 𝑦𝑅𝑧
74, 6nfralw 3302 . . . . 5 𝑥𝑦𝐵 ¬ 𝑦𝑅𝑧
85, 7nfan 1894 . . . 4 𝑥(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)
9 eleq1w 2810 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
10 breq2 5145 . . . . . . 7 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
1110notbid 318 . . . . . 6 (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
1211ralbidv 3171 . . . . 5 (𝑥 = 𝑧 → (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
139, 12anbi12d 630 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥) ↔ (𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)))
142, 8, 13cbvexv1 2332 . . 3 (∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥) ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
15 df-rex 3065 . . 3 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
16 df-rex 3065 . . 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 395  w3a 1084  wal 1531  wex 1773  wcel 2098  wne 2934  wral 3055  wrex 3064  wss 3943  c0 4317   class class class wbr 5141   FrSe w-bnj15 34231
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-reg 9586  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7852  df-1o 8464  df-bnj17 34226  df-bnj14 34228  df-bnj13 34230  df-bnj15 34232  df-bnj18 34234  df-bnj19 34236
This theorem is referenced by:  bnj1204  34551  bnj1311  34563  bnj1312  34597
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