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Theorem onminesb 7813
Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses explicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 29-Sep-2003.)
Assertion
Ref Expression
onminesb (∃𝑥 ∈ On 𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)

Proof of Theorem onminesb
StepHypRef Expression
1 rabn0 4395 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
2 ssrab2 4090 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
3 onint 7810 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
42, 3mpan 690 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
51, 4sylbir 235 . 2 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑})
6 nfcv 2903 . . . 4 𝑥On
76elrabsf 3840 . . 3 ( {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ ( {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ [ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
87simprbi 496 . 2 ( {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} → [ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
95, 8syl 17 1 (∃𝑥 ∈ On 𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2938  wrex 3068  {crab 3433  [wsbc 3791  wss 3963  c0 4339   cint 4951  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  onminex  7822
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