Theorem bnj1228 34989

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	⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)

Assertion

Ref	Expression
bnj1228	⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 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.

Step	Hyp	Ref	Expression
1		bnj69 34988	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
2		nfv 1913	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
3		bnj1228.1	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
4	3	nfcii 2897	. . . . . 6 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2900	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
6		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑦𝑅𝑧
7	4, 6	nfralw 3317	. . . . 5 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧
8	5, 7	nfan 1898	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
9		eleq1w 2827	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
10		breq2 5170	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
11	10	notbid 318	. . . . . 6 ⊢ (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
12	11	ralbidv 3184	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧))
13	9, 12	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)))
14	2, 8, 13	cbvexv1 2348	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧))
15		df-rex 3077	. . 3 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
16		df-rex 3077	. . 3 ⊢ (∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧))
17	14, 15, 16	3bitr4i 303	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
18	1, 17	sylibr 234	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1535  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166   FrSe w-bnj15 34670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-reg 9663  ax-inf2 9712

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-om 7906  df-1o 8524  df-bnj17 34665  df-bnj14 34667  df-bnj13 34669  df-bnj15 34671  df-bnj18 34673  df-bnj19 34675

This theorem is referenced by:  bnj1204  34990  bnj1311  35002  bnj1312  35036