Theorem onminex 2275

Description: If a wff is true for an ordinal number, there is a smallest ordinal number for which it is true.

Hypothesis

Ref	Expression
onminex.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
onminex	⊢ (∃x ∈ On φ → ∃x ∈ On (φ ∧ ∀y ∈ x ¬ ψ))

Distinct variable group(s):   x,y   φ,y   ψ,x

Proof of Theorem onminex

Step	Hyp	Ref	Expression
1		hbab1 1095	. . . . . . 7 ⊢ (y ∈ {x∣(x ∈ On ∧ φ)} → ∀x y ∈ {x∣(x ∈ On ∧ φ)})
2	1	hbint 1975	. . . . . 6 ⊢ (y ∈ ∩{x∣(x ∈ On ∧ φ)} → ∀x y ∈ ∩{x∣(x ∈ On ∧ φ)})
3	2, 1	hbel 1172	. . . . . . 7 ⊢ (∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)} → ∀x∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)})
4		ax-17 925	. . . . . . . . 9 ⊢ (¬ ψ → ∀x ¬ ψ)
5	2, 4	hbim 702	. . . . . . . 8 ⊢ ((y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ) → ∀x(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ))
6	5	hbal 700	. . . . . . 7 ⊢ (∀y(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ) → ∀x∀y(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ))
7	3, 6	hban 704	. . . . . 6 ⊢ ((∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ)) → ∀x(∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ)))
8		eleq1 1149	. . . . . . 7 ⊢ (x = ∩{x∣(x ∈ On ∧ φ)} → (x ∈ {x∣(x ∈ On ∧ φ)} ↔ ∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)}))
9		eleq2 1150	. . . . . . . . 9 ⊢ (x = ∩{x∣(x ∈ On ∧ φ)} → (y ∈ x ↔ y ∈ ∩{x∣(x ∈ On ∧ φ)}))
10	9	imbi1d 465	. . . . . . . 8 ⊢ (x = ∩{x∣(x ∈ On ∧ φ)} → ((y ∈ x → ¬ ψ) ↔ (y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ)))
11	10	bialdv 935	. . . . . . 7 ⊢ (x = ∩{x∣(x ∈ On ∧ φ)} → (∀y(y ∈ x → ¬ ψ) ↔ ∀y(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ)))
12	8, 11	anbi12d 476	. . . . . 6 ⊢ (x = ∩{x∣(x ∈ On ∧ φ)} → ((x ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ x → ¬ ψ)) ↔ (∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ))))
13	2, 7, 12	cla4egf 1395	. . . . 5 ⊢ (∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)} → ((∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ)) → ∃x(x ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ x → ¬ ψ))))
14	13	anabsi5 377	. . . 4 ⊢ ((∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ)) → ∃x(x ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ x → ¬ ψ)))
15		ssab 1555	. . . . 5 ⊢ {x∣(x ∈ On ∧ φ)} ⊆ On
16		onint 2261	. . . . 5 ⊢ (({x∣(x ∈ On ∧ φ)} ⊆ On ∧ ¬ {x∣(x ∈ On ∧ φ)} = ∅) → ∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)})
17	15, 16	mpan 518	. . . 4 ⊢ (¬ {x∣(x ∈ On ∧ φ)} = ∅ → ∩{x∣(x ∈ On ∧ φ)} ∈ {x∣(x ∈ On ∧ φ)})
18		oninton 2267	. . . . . . . 8 ⊢ (({x∣(x ∈ On ∧ φ)} ⊆ On ∧ ¬ {x∣(x ∈ On ∧ φ)} = ∅) → ∩{x∣(x ∈ On ∧ φ)} ∈ On)
19	15, 18	mpan 518	. . . . . . 7 ⊢ (¬ {x∣(x ∈ On ∧ φ)} = ∅ → ∩{x∣(x ∈ On ∧ φ)} ∈ On)
20		onelon 2223	. . . . . . . 8 ⊢ ((∩{x∣(x ∈ On ∧ φ)} ∈ On ∧ y ∈ ∩{x∣(x ∈ On ∧ φ)}) → y ∈ On)
21	20	exp 291	. . . . . . 7 ⊢ (∩{x∣(x ∈ On ∧ φ)} ∈ On → (y ∈ ∩{x∣(x ∈ On ∧ φ)} → y ∈ On))
22	19, 21	syl 12	. . . . . 6 ⊢ (¬ {x∣(x ∈ On ∧ φ)} = ∅ → (y ∈ ∩{x∣(x ∈ On ∧ φ)} → y ∈ On))
23		visset 1350	. . . . . . . . . 10 ⊢ y ∈ V
24		eleq1 1149	. . . . . . . . . . 11 ⊢ (x = y → (x ∈ On ↔ y ∈ On))
25		onminex.1	. . . . . . . . . . 11 ⊢ (x = y → (φ ↔ ψ))
26	24, 25	anbi12d 476	. . . . . . . . . 10 ⊢ (x = y → ((x ∈ On ∧ φ) ↔ (y ∈ On ∧ ψ)))
27	23, 26	elab 1415	. . . . . . . . 9 ⊢ (y ∈ {x∣(x ∈ On ∧ φ)} ↔ (y ∈ On ∧ ψ))
28		onnmin 2270	. . . . . . . . . 10 ⊢ (({x∣(x ∈ On ∧ φ)} ⊆ On ∧ y ∈ {x∣(x ∈ On ∧ φ)}) → ¬ y ∈ ∩{x∣(x ∈ On ∧ φ)})
29	15, 28	mpan 518	. . . . . . . . 9 ⊢ (y ∈ {x∣(x ∈ On ∧ φ)} → ¬ y ∈ ∩{x∣(x ∈ On ∧ φ)})
30	27, 29	sylbir 176	. . . . . . . 8 ⊢ ((y ∈ On ∧ ψ) → ¬ y ∈ ∩{x∣(x ∈ On ∧ φ)})
31	30	exp 291	. . . . . . 7 ⊢ (y ∈ On → (ψ → ¬ y ∈ ∩{x∣(x ∈ On ∧ φ)}))
32	31	con2d 83	. . . . . 6 ⊢ (y ∈ On → (y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ))
33	22, 32	syli 52	. . . . 5 ⊢ (¬ {x∣(x ∈ On ∧ φ)} = ∅ → (y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ))
34	33	19.21aiv 943	. . . 4 ⊢ (¬ {x∣(x ∈ On ∧ φ)} = ∅ → ∀y(y ∈ ∩{x∣(x ∈ On ∧ φ)} → ¬ ψ))
35	14, 17, 34	sylanc 361	. . 3 ⊢ (¬ {x∣(x ∈ On ∧ φ)} = ∅ → ∃x(x ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ x → ¬ ψ)))
36		abn0 1715	. . 3 ⊢ (¬ {x∣(x ∈ On ∧ φ)} = ∅ ↔ ∃x(x ∈ On ∧ φ))
37		abid 1094	. . . . . . 7 ⊢ (x ∈ {x∣(x ∈ On ∧ φ)} ↔ (x ∈ On ∧ φ))
38	37	bicomi 150	. . . . . 6 ⊢ ((x ∈ On ∧ φ) ↔ x ∈ {x∣(x ∈ On ∧ φ)})
39		df-ral 1205	. . . . . 6 ⊢ (∀y ∈ x ¬ ψ ↔ ∀y(y ∈ x → ¬ ψ))
40	38, 39	anbi12i 369	. . . . 5 ⊢ (((x ∈ On ∧ φ) ∧ ∀y ∈ x ¬ ψ) ↔ (x ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ x → ¬ ψ)))
41		anass 336	. . . . 5 ⊢ (((x ∈ On ∧ φ) ∧ ∀y ∈ x ¬ ψ) ↔ (x ∈ On ∧ (φ ∧ ∀y ∈ x ¬ ψ)))
42	40, 41	bitr3 153	. . . 4 ⊢ ((x ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ x → ¬ ψ)) ↔ (x ∈ On ∧ (φ ∧ ∀y ∈ x ¬ ψ)))
43	42	biex 733	. . 3 ⊢ (∃x(x ∈ {x∣(x ∈ On ∧ φ)} ∧ ∀y(y ∈ x → ¬ ψ)) ↔ ∃x(x ∈ On ∧ (φ ∧ ∀y ∈ x ¬ ψ)))
44	35, 36, 43	3imtr3 191	. 2 ⊢ (∃x(x ∈ On ∧ φ) → ∃x(x ∈ On ∧ (φ ∧ ∀y ∈ x ¬ ψ)))
45		df-rex 1206	. 2 ⊢ (∃x ∈ On φ ↔ ∃x(x ∈ On ∧ φ))
46		df-rex 1206	. 2 ⊢ (∃x ∈ On (φ ∧ ∀y ∈ x ¬ ψ) ↔ ∃x(x ∈ On ∧ (φ ∧ ∀y ∈ x ¬ ψ)))
47	44, 45, 46	3imtr4 192	1 ⊢ (∃x ∈ On φ → ∃x ∈ On (φ ∧ ∀y ∈ x ¬ ψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  ∅c0 1707  ∩cint 1965  Oncon0 2199

This theorem is referenced by:  tz7.49 2997  zornlem7 3609

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-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

metamath.org