Theorem bnj1228 34017

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 34016	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
2		nfv 1917	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
3		bnj1228.1	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
4	3	nfcii 2887	. . . . . 6 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2890	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
6		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑦𝑅𝑧
7	4, 6	nfralw 3308	. . . . 5 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧
8	5, 7	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
9		eleq1w 2816	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
10		breq2 5152	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
11	10	notbid 317	. . . . . 6 ⊢ (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
12	11	ralbidv 3177	. . . . 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 2338	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧))
15		df-rex 3071	. . 3 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
16		df-rex 3071	. . 3 ⊢ (∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧))
17	14, 15, 16	3bitr4i 302	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
18	1, 17	sylibr 233	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087  ∀wal 1539  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148   FrSe w-bnj15 33698

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-reg 9586  ax-inf2 9635

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-bnj17 33693  df-bnj14 33695  df-bnj13 33697  df-bnj15 33699  df-bnj18 33701  df-bnj19 33703

This theorem is referenced by:  bnj1204  34018  bnj1311  34030  bnj1312  34064