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Theorem bnj1228 34675
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 34674 . 2 ((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
2 nfv 1909 . . . 4 𝑧(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
3 bnj1228.1 . . . . . . 7 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
43nfcii 2883 . . . . . 6 𝑥𝐵
54nfcri 2886 . . . . 5 𝑥 𝑧𝐵
6 nfv 1909 . . . . . 6 𝑥 ¬ 𝑦𝑅𝑧
74, 6nfralw 3306 . . . . 5 𝑥𝑦𝐵 ¬ 𝑦𝑅𝑧
85, 7nfan 1894 . . . 4 𝑥(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)
9 eleq1w 2812 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
10 breq2 5156 . . . . . . 7 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
1110notbid 317 . . . . . 6 (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
1211ralbidv 3175 . . . . 5 (𝑥 = 𝑧 → (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
139, 12anbi12d 630 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥) ↔ (𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)))
142, 8, 13cbvexv1 2333 . . 3 (∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥) ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
15 df-rex 3068 . . 3 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
16 df-rex 3068 . . 3 (∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
1714, 15, 163bitr4i 302 . 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 394  w3a 1084  wal 1531  wex 1773  wcel 2098  wne 2937  wral 3058  wrex 3067  wss 3949  c0 4326   class class class wbr 5152   FrSe w-bnj15 34356
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-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9623  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-1o 8493  df-bnj17 34351  df-bnj14 34353  df-bnj13 34355  df-bnj15 34357  df-bnj18 34359  df-bnj19 34361
This theorem is referenced by:  bnj1204  34676  bnj1311  34688  bnj1312  34722
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