Theorem onminex 6899

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 → (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 3650	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		rabn0 3912	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
3	2	biimpri 217	. . . 4 ⊢ (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4		oninton 6892	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
5	1, 3, 4	sylancr 694	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
6		onminesb 6890	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7		onss 6882	. . . . . . 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 3567	. . . . 5 ⊢ (∃𝑥 ∈ On 𝜑 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10		onminex.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	10	onnminsb 6896	. . . . 5 ⊢ (𝑦 ∈ On → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
12	9, 11	syli 38	. . . 4 ⊢ (∃𝑥 ∈ On 𝜑 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
13	12	ralrimiv 2948	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)
14		dfsbcq2 3405	. . . . 5 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → ([𝑧 / 𝑥]𝜑 ↔ [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
15		raleq 3115	. . . . 5 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (∀𝑦 ∈ 𝑧 ¬ 𝜓 ↔ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓))
16	14, 15	anbi12d 743	. . . 4 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓) ↔ ([∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
17	16	rspcev 3282	. . 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 1316	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
19		nfv 1830	. . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓)
20		nfs1v 2425	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
21		nfv 1830	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 ¬ 𝜓
22	20, 21	nfan 1816	. . 3 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)
23		sbequ12 2097	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24		raleq 3115	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
25	23, 24	anbi12d 743	. . 3 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)))
26	19, 22, 25	cbvrex 3144	. 2 ⊢ (∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
27	18, 26	sylibr 223	1 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  [wsb 1867   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  [wsbc 3402   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644

This theorem is referenced by:  tz7.49  7427  omeulem1  7549  zorn2lem7  9207