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Theorem onminex 6954
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 3666 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 rabn0 3932 . . . . 5 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
32biimpri 218 . . . 4 (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4 oninton 6947 . . . 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 6945 . . 3 (∃𝑥 ∈ On 𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7 onss 6937 . . . . . . 7 ( {𝑥 ∈ On ∣ 𝜑} ∈ On → {𝑥 ∈ On ∣ 𝜑} ⊆ On)
85, 7syl 17 . . . . . 6 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ⊆ On)
98sseld 3582 . . . . 5 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10 onminex.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
1110onnminsb 6951 . . . . 5 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
129, 11syli 39 . . . 4 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
1312ralrimiv 2959 . . 3 (∃𝑥 ∈ On 𝜑 → ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)
14 dfsbcq2 3420 . . . . 5 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → ([𝑧 / 𝑥]𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
15 raleq 3127 . . . . 5 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → (∀𝑦𝑧 ¬ 𝜓 ↔ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓))
1614, 15anbi12d 746 . . . 4 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓) ↔ ([ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
1716rspcev 3295 . . 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 1321 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
19 nfv 1840 . . 3 𝑧(𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓)
20 nfs1v 2436 . . . 4 𝑥[𝑧 / 𝑥]𝜑
21 nfv 1840 . . . 4 𝑥𝑦𝑧 ¬ 𝜓
2220, 21nfan 1825 . . 3 𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)
23 sbequ12 2108 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24 raleq 3127 . . . 4 (𝑥 = 𝑧 → (∀𝑦𝑥 ¬ 𝜓 ↔ ∀𝑦𝑧 ¬ 𝜓))
2523, 24anbi12d 746 . . 3 (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)))
2619, 22, 25cbvrex 3156 . 2 (∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓) ↔ ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
2718, 26sylibr 224 1 (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  [wsb 1877  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  [wsbc 3417  wss 3555  c0 3891   cint 4440  Oncon0 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686
This theorem is referenced by:  tz7.49  7485  omeulem1  7607  zorn2lem7  9268
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