Theorem onminex 7822

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 4080	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		rabn0 4389	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
3	2	biimpri 228	. . . 4 ⊢ (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4		oninton 7815	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
5	1, 3, 4	sylancr 587	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
6		onminesb 7813	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7		onss 7805	. . . . . . 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 3982	. . . . 5 ⊢ (∃𝑥 ∈ On 𝜑 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10		onminex.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	10	onnminsb 7819	. . . . 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 3145	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)
14		dfsbcq2 3791	. . . . 5 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → ([𝑧 / 𝑥]𝜑 ↔ [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
15		raleq 3323	. . . . 5 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (∀𝑦 ∈ 𝑧 ¬ 𝜓 ↔ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓))
16	14, 15	anbi12d 632	. . . 4 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓) ↔ ([∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
17	16	rspcev 3622	. . 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 837	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
19		nfv 1914	. . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓)
20		nfs1v 2156	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
21		nfv 1914	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 ¬ 𝜓
22	20, 21	nfan 1899	. . 3 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)
23		sbequ12 2251	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24		raleq 3323	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
25	23, 24	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)))
26	19, 22, 25	cbvrexw 3307	. 2 ⊢ (∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
27	18, 26	sylibr 234	1 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  [wsb 2064   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436  [wsbc 3788   ⊆ wss 3951  ∅c0 4333  ∩ cint 4946  Oncon0 6384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388

This theorem is referenced by:  tz7.49  8485  omeulem1  8620  zorn2lem7  10542