Theorem onnminsb 2271

Description: An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. ψ is the wff resulting from the substitution of A for x in wff φ.

Hypothesis

Ref	Expression
onnminsb.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
onnminsb	⊢ (A ∈ On → (A ∈ ∩{x ∈ On∣φ} → ¬ ψ))

Distinct variable group(s):   x,A   ψ,x

Proof of Theorem onnminsb

Step	Hyp	Ref	Expression
1		onnminsb.1	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
2	1	elrab 1422	. . . 4 ⊢ (A ∈ {x ∈ On∣φ} ↔ (A ∈ On ∧ ψ))
3		ssrab 1556	. . . . 5 ⊢ {x ∈ On∣φ} ⊆ On
4		onnmin 2270	. . . . 5 ⊢ (({x ∈ On∣φ} ⊆ On ∧ A ∈ {x ∈ On∣φ}) → ¬ A ∈ ∩{x ∈ On∣φ})
5	3, 4	mpan 518	. . . 4 ⊢ (A ∈ {x ∈ On∣φ} → ¬ A ∈ ∩{x ∈ On∣φ})
6	2, 5	sylbir 176	. . 3 ⊢ ((A ∈ On ∧ ψ) → ¬ A ∈ ∩{x ∈ On∣φ})
7	6	exp 291	. 2 ⊢ (A ∈ On → (ψ → ¬ A ∈ ∩{x ∈ On∣φ}))
8	7	con2d 83	1 ⊢ (A ∈ On → (A ∈ ∩{x ∈ On∣φ} → ¬ ψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {crab 1204   ⊆ wss 1487  ∩cint 1965  Oncon0 2199

This theorem is referenced by:  oawordeulem 3156  alephnbtwn 3674  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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