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Theorem onminex 7240
Description: If a wff is true for an ordinal number, then 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 3891 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 rabn0 4165 . . . . 5 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
32biimpri 219 . . . 4 (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4 oninton 7233 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
51, 3, 4sylancr 577 . . 3 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
6 onminesb 7231 . . 3 (∃𝑥 ∈ On 𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7 onss 7223 . . . . . . 7 ( {𝑥 ∈ On ∣ 𝜑} ∈ On → {𝑥 ∈ On ∣ 𝜑} ⊆ On)
85, 7syl 17 . . . . . 6 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ⊆ On)
98sseld 3804 . . . . 5 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10 onminex.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
1110onnminsb 7237 . . . . 5 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
129, 11syli 39 . . . 4 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
1312ralrimiv 3160 . . 3 (∃𝑥 ∈ On 𝜑 → ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)
14 dfsbcq2 3643 . . . . 5 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → ([𝑧 / 𝑥]𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
15 raleq 3334 . . . . 5 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → (∀𝑦𝑧 ¬ 𝜓 ↔ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓))
1614, 15anbi12d 618 . . . 4 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓) ↔ ([ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
1716rspcev 3509 . . 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 856 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
19 nfv 2005 . . 3 𝑧(𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓)
20 nfs1v 2288 . . . 4 𝑥[𝑧 / 𝑥]𝜑
21 nfv 2005 . . . 4 𝑥𝑦𝑧 ¬ 𝜓
2220, 21nfan 1990 . . 3 𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)
23 sbequ12 2280 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24 raleq 3334 . . . 4 (𝑥 = 𝑧 → (∀𝑦𝑥 ¬ 𝜓 ↔ ∀𝑦𝑧 ¬ 𝜓))
2523, 24anbi12d 618 . . 3 (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)))
2619, 22, 25cbvrex 3364 . 2 (∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓) ↔ ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
2718, 26sylibr 225 1 (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  [wsb 2061  wcel 2157  wne 2985  wral 3103  wrex 3104  {crab 3107  [wsbc 3640  wss 3776  c0 4123   cint 4676  Oncon0 5943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-br 4852  df-opab 4914  df-tr 4954  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-ord 5946  df-on 5947
This theorem is referenced by:  tz7.49  7779  omeulem1  7902  zorn2lem7  9612
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