Theorem onnminsb 7789

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. (Contributed by NM, 9-Nov-2003.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem onnminsb

Step	Hyp	Ref	Expression
1		onnminsb.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	elrab 3682	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓))
3		ssrab2 4076	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
4		onnmin 7788	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑})
5	3, 4	mpan 686	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑})
6	2, 5	sylbir 234	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝜓) → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑})
7	6	ex 411	. 2 ⊢ (𝐴 ∈ On → (𝜓 → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑}))
8	7	con2d 134	1 ⊢ (𝐴 ∈ On → (𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {crab 3430   ⊆ wss 3947  ∩ cint 4949  Oncon0 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366  df-on 6367

This theorem is referenced by:  onminex  7792  oawordeulem  8556  oeeulem  8603  nnawordex  8639  tcrank  9881  alephnbtwn  10068  cardaleph  10086  cardmin  10561  sltval2  27395  nosepeq  27424  nosupbnd2lem1  27454  noinfbnd2lem1  27469  naddwordnexlem4  42454