Theorem onnminsb 7798

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 3659	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓))
3		ssrab2 4042	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
4		onnmin 7797	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑})
5	3, 4	mpan 702	. . . 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 417	. 2 ⊢ (𝐴 ∈ On → (𝜓 → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑}))
8	7	con2d 135	1 ⊢ (𝐴 ∈ On → (𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423   ⊆ wss 3913  ∩ cint 4916  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365

This theorem is referenced by:  onminex  7801  oawordeulem  8539  oeeulem  8587  nnawordex  8623  tcrank  9856  alephnbtwn  10055  cardaleph  10073  cardmin  10548  ltsval2  27786  nosepeq  27815  nosupbnd2lem1  27845  noinfbnd2lem1  27860  onvf1odlem4  35489  naddwordnexlem4  44020