Theorem onminsb 2264

Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228.

Hypotheses

Ref	Expression
onminsb.1	⊢ (ψ → ∀xψ)
onminsb.2	⊢ (x = ∩{x ∈ On∣φ} → (φ ↔ ψ))

Assertion

Ref	Expression
onminsb	⊢ (∃x ∈ On φ → ψ)

Proof of Theorem onminsb

Step	Hyp	Ref	Expression
1		rabn0 1716	. . 3 ⊢ (¬ {x ∈ On∣φ} = ∅ ↔ ∃x ∈ On φ)
2		ssrab 1556	. . . 4 ⊢ {x ∈ On∣φ} ⊆ On
3		onint 2261	. . . 4 ⊢ (({x ∈ On∣φ} ⊆ On ∧ ¬ {x ∈ On∣φ} = ∅) → ∩{x ∈ On∣φ} ∈ {x ∈ On∣φ})
4	2, 3	mpan 518	. . 3 ⊢ (¬ {x ∈ On∣φ} = ∅ → ∩{x ∈ On∣φ} ∈ {x ∈ On∣φ})
5	1, 4	sylbir 176	. 2 ⊢ (∃x ∈ On φ → ∩{x ∈ On∣φ} ∈ {x ∈ On∣φ})
6		hbrab1 1310	. . . . 5 ⊢ (y ∈ {x ∈ On∣φ} → ∀x y ∈ {x ∈ On∣φ})
7	6	hbint 1975	. . . 4 ⊢ (y ∈ ∩{x ∈ On∣φ} → ∀x y ∈ ∩{x ∈ On∣φ})
8		ax-17 925	. . . 4 ⊢ (y ∈ On → ∀x y ∈ On)
9		onminsb.1	. . . 4 ⊢ (ψ → ∀xψ)
10		onminsb.2	. . . 4 ⊢ (x = ∩{x ∈ On∣φ} → (φ ↔ ψ))
11	7, 8, 9, 10	elrabf 1421	. . 3 ⊢ (∩{x ∈ On∣φ} ∈ {x ∈ On∣φ} ↔ (∩{x ∈ On∣φ} ∈ On ∧ ψ))
12	11	pm3.27bd 263	. 2 ⊢ (∩{x ∈ On∣φ} ∈ {x ∈ On∣φ} → ψ)
13	5, 12	syl 12	1 ⊢ (∃x ∈ On φ → ψ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204   ⊆ wss 1487  ∅c0 1707  ∩cint 1965  Oncon0 2199

This theorem is referenced by:  oawordeulem 3156  rankid 3516  cardmin 3666  alephordlem1 3677  cardaleph 3690

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-rab 1208  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

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