Theorem tz9.12 3506

Description: A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 3503 through tz9.12lem3 3505.

Hypothesis

Ref	Expression
tz9.12.1	⊢ A ∈ V

Assertion

Ref	Expression
tz9.12	⊢ (∀x ∈ A ∃y ∈ On x ∈ (R1 ‘y) → ∃y ∈ On A ∈ (R1 ‘y))

Distinct variable group(s):   x,y,A

Proof of Theorem tz9.12

Step	Hyp	Ref	Expression
1		tz9.12.1	. . 3 ⊢ A ∈ V
2		cleqid 1102	. . 3 ⊢ {⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} = {⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}}
3	1, 2	tz9.12lem3 3505	. 2 ⊢ (∀x ∈ A ∃y ∈ On x ∈ (R1 ‘y) → A ∈ (R1 ‘suc suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A)))
4	1, 2	tz9.12lem2 3504	. . . 4 ⊢ suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A) ∈ On
5	4	onsuc 2353	. . 3 ⊢ suc suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A) ∈ On
6		fveq2 2832	. . . . 5 ⊢ (y = suc suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A) → (R1 ‘y) = (R1 ‘suc suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A)))
7	6	eleq2d 1156	. . . 4 ⊢ (y = suc suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A) → (A ∈ (R1 ‘y) ↔ A ∈ (R1 ‘suc suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A))))
8	7	rcla4ev 1403	. . 3 ⊢ ((suc suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A) ∈ On ∧ A ∈ (R1 ‘suc suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A))) → ∃y ∈ On A ∈ (R1 ‘y))
9	5, 8	mpan 518	. 2 ⊢ (A ∈ (R1 ‘suc suc ∪({⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} “ A)) → ∃y ∈ On A ∈ (R1 ‘y))
10	3, 9	syl 12	1 ⊢ (∀x ∈ A ∃y ∈ On x ∈ (R1 ‘y) → ∃y ∈ On A ∈ (R1 ‘y))

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348  ∪cuni 1919  ∩cint 1965  {copab 2055  Oncon0 2199  suc csuc 2201   “ cima 2413   ‘cfv 2422  R₁cr1 3485

This theorem is referenced by:  tz9.13 3507

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-r1 3487

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