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Theorem onminex 7742
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 4042 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 rabn0 4350 . . . . 5 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
32biimpri 227 . . . 4 (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4 oninton 7735 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
51, 3, 4sylancr 588 . . 3 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
6 onminesb 7733 . . 3 (∃𝑥 ∈ On 𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7 onss 7724 . . . . . . 7 ( {𝑥 ∈ On ∣ 𝜑} ∈ On → {𝑥 ∈ On ∣ 𝜑} ⊆ On)
85, 7syl 17 . . . . . 6 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ⊆ On)
98sseld 3948 . . . . 5 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10 onminex.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
1110onnminsb 7739 . . . . 5 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
129, 11syli 39 . . . 4 (∃𝑥 ∈ On 𝜑 → (𝑦 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
1312ralrimiv 3143 . . 3 (∃𝑥 ∈ On 𝜑 → ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)
14 dfsbcq2 3747 . . . . 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 632 . . . 4 (𝑧 = {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓) ↔ ([ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
1716rspcev 3584 . . 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 836 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓))
19 nfv 1918 . . 3 𝑧(𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓)
20 nfs1v 2154 . . . 4 𝑥[𝑧 / 𝑥]𝜑
21 nfv 1918 . . . 4 𝑥𝑦𝑧 ¬ 𝜓
2220, 21nfan 1903 . . 3 𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)
23 sbequ12 2244 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24 raleq 3312 . . . 4 (𝑥 = 𝑧 → (∀𝑦𝑥 ¬ 𝜓 ↔ ∀𝑦𝑧 ¬ 𝜓))
2523, 24anbi12d 632 . . 3 (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦𝑧 ¬ 𝜓)))
2619, 22, 25cbvrexw 3293 . 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 397   = wceq 1542  [wsb 2068  wcel 2107  wne 2944  wral 3065  wrex 3074  {crab 3410  [wsbc 3744  wss 3915  c0 4287   cint 4912  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  tz7.49  8396  omeulem1  8534  zorn2lem7  10445
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