Theorem onminex 7007

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 3687	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		rabn0 3958	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
3	2	biimpri 218	. . . 4 ⊢ (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4		oninton 7000	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
5	1, 3, 4	sylancr 695	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
6		onminesb 6998	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7		onss 6990	. . . . . . 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 3602	. . . . 5 ⊢ (∃𝑥 ∈ On 𝜑 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10		onminex.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	10	onnminsb 7004	. . . . 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 2965	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)
14		dfsbcq2 3438	. . . . 5 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → ([𝑧 / 𝑥]𝜑 ↔ [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑))
15		raleq 3138	. . . . 5 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (∀𝑦 ∈ 𝑧 ¬ 𝜓 ↔ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓))
16	14, 15	anbi12d 747	. . . 4 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓) ↔ ([∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
17	16	rspcev 3309	. . 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 1324	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
19		nfv 1843	. . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓)
20		nfs1v 2437	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
21		nfv 1843	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 ¬ 𝜓
22	20, 21	nfan 1828	. . 3 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)
23		sbequ12 2111	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24		raleq 3138	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
25	23, 24	anbi12d 747	. . 3 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)))
26	19, 22, 25	cbvrex 3168	. 2 ⊢ (∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
27	18, 26	sylibr 224	1 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  [wsb 1880   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  [wsbc 3435   ⊆ wss 3574  ∅c0 3915  ∩ cint 4475  Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  tz7.49  7540  omeulem1  7662  zorn2lem7  9324