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Theorem onminex 7811
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 4076 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 rabn0 4390 . . . . 5 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
32biimpri 227 . . . 4 (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4 oninton 7804 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
51, 3, 4sylancr 585 . . 3 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
6 onminesb 7802 . . 3 (∃𝑥 ∈ On 𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7 onss 7793 . . . . . . 7 ( {𝑥 ∈ On ∣ 𝜑} ∈ On → {𝑥 ∈ On ∣ 𝜑} ⊆ On)
85, 7syl 17 . . . . . 6 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ⊆ On)
98sseld 3978 . . . . 5 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10 onminex.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
1110onnminsb 7808 . . . . 5 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
129, 11syli 39 . . . 4 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
1312ralrimiv 3135 . . 3 (∃𝑥 ∈ On 𝜑 → ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)
14 dfsbcq2 3779 . . . . 5 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → ([𝑧 / 𝑥]𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
15 raleq 3312 . . . . 5 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → (∀𝑦𝑧 ¬ 𝜓 ↔ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓))
1614, 15anbi12d 630 . . . 4 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓) ↔ ([ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
1716rspcev 3608 . . 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 835 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
19 nfv 1910 . . 3 𝑧(𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓)
20 nfs1v 2146 . . . 4 𝑥[𝑧 / 𝑥]𝜑
21 nfv 1910 . . . 4 𝑥𝑦𝑧 ¬ 𝜓
2220, 21nfan 1895 . . 3 𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)
23 sbequ12 2239 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24 raleq 3312 . . . 4 (𝑥 = 𝑧 → (∀𝑦𝑥 ¬ 𝜓 ↔ ∀𝑦𝑧 ¬ 𝜓))
2523, 24anbi12d 630 . . 3 (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)))
2619, 22, 25cbvrexw 3295 . 2 (∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓) ↔ ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
2718, 26sylibr 233 1 (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  [wsb 2060  wcel 2099  wne 2930  wral 3051  wrex 3060  {crab 3419  [wsbc 3776  wss 3947  c0 4325   cint 4954  Oncon0 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380
This theorem is referenced by:  tz7.49  8475  omeulem1  8612  zorn2lem7  10545
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