Theorem onminex 7629

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 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
onminex	⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 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.

Step	Hyp	Ref	Expression
1		ssrab2 4009	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		rabn0 4316	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
3	2	biimpri 227	. . . 4 ⊢ (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4		oninton 7622	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
5	1, 3, 4	sylancr 586	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
6		onminesb 7620	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7		onss 7611	. . . . . . 7 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ On)
8	5, 7	syl 17	. . . . . 6 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ On)
9	8	sseld 3916	. . . . 5 ⊢ (∃𝑥 ∈ On 𝜑 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10		onminex.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	10	onnminsb 7626	. . . . 5 ⊢ (𝑦 ∈ On → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
12	9, 11	syli 39	. . . 4 ⊢ (∃𝑥 ∈ On 𝜑 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
13	12	ralrimiv 3106	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)
14		dfsbcq2 3714	. . . . 5 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → ([𝑧 / 𝑥]𝜑 ↔ [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
15		raleq 3333	. . . . 5 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (∀𝑦 ∈ 𝑧 ¬ 𝜓 ↔ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓))
16	14, 15	anbi12d 630	. . . 4 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓) ↔ ([∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
17	16	rspcev 3552	. . 3 ⊢ ((∩ {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ ([∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)) → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
18	5, 6, 13, 17	syl12anc 833	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
19		nfv 1918	. . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓)
20		nfs1v 2155	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
21		nfv 1918	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 ¬ 𝜓
22	20, 21	nfan 1903	. . 3 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)
23		sbequ12 2247	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24		raleq 3333	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
25	23, 24	anbi12d 630	. . 3 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)))
26	19, 22, 25	cbvrexw 3364	. 2 ⊢ (∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
27	18, 26	sylibr 233	1 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  [wsb 2068   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  [wsbc 3711   ⊆ wss 3883  ∅c0 4253  ∩ cint 4876  Oncon0 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255

This theorem is referenced by:  tz7.49  8246  omeulem1  8375  zorn2lem7  10189