Theorem onnminsb 7499

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 3628	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓))
3		ssrab2 4007	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
4		onnmin 7498	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑})
5	3, 4	mpan 689	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑})
6	2, 5	sylbir 238	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝜓) → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑})
7	6	ex 416	. 2 ⊢ (𝐴 ∈ On → (𝜓 → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑}))
8	7	con2d 136	1 ⊢ (𝐴 ∈ On → (𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110   ⊆ wss 3881  ∩ cint 4838  Oncon0 6159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163

This theorem is referenced by:  onminex  7502  oawordeulem  8163  oeeulem  8210  nnawordex  8246  tcrank  9297  alephnbtwn  9482  cardaleph  9500  cardmin  9975  sltval2  33276  nosepeq  33302  nosupbnd2lem1  33328