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Theorem onminex 6899
 Description: If a wff is true for an ordinal number, there is the smallest ordinal number for which it is true. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Mario Carneiro, 20-Nov-2016.)
Hypothesis
Ref Expression
onminex.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
onminex (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem onminex
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3650 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 rabn0 3912 . . . . 5 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
32biimpri 217 . . . 4 (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4 oninton 6892 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
51, 3, 4sylancr 694 . . 3 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
6 onminesb 6890 . . 3 (∃𝑥 ∈ On 𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7 onss 6882 . . . . . . 7 ( {𝑥 ∈ On ∣ 𝜑} ∈ On → {𝑥 ∈ On ∣ 𝜑} ⊆ On)
85, 7syl 17 . . . . . 6 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ⊆ On)
98sseld 3567 . . . . 5 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10 onminex.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
1110onnminsb 6896 . . . . 5 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
129, 11syli 38 . . . 4 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
1312ralrimiv 2948 . . 3 (∃𝑥 ∈ On 𝜑 → ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)
14 dfsbcq2 3405 . . . . 5 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → ([𝑧 / 𝑥]𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
15 raleq 3115 . . . . 5 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → (∀𝑦𝑧 ¬ 𝜓 ↔ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓))
1614, 15anbi12d 743 . . . 4 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓) ↔ ([ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
1716rspcev 3282 . . 3 (( {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ ([ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)) → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
185, 6, 13, 17syl12anc 1316 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
19 nfv 1830 . . 3 𝑧(𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓)
20 nfs1v 2425 . . . 4 𝑥[𝑧 / 𝑥]𝜑
21 nfv 1830 . . . 4 𝑥𝑦𝑧 ¬ 𝜓
2220, 21nfan 1816 . . 3 𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)
23 sbequ12 2097 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24 raleq 3115 . . . 4 (𝑥 = 𝑧 → (∀𝑦𝑥 ¬ 𝜓 ↔ ∀𝑦𝑧 ¬ 𝜓))
2523, 24anbi12d 743 . . 3 (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)))
2619, 22, 25cbvrex 3144 . 2 (∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓) ↔ ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
2718, 26sylibr 223 1 (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  [wsb 1867   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  [wsbc 3402   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by:  tz7.49  7427  omeulem1  7549  zorn2lem7  9207
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