Theorem onminex 7781

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 4033	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		rabn0 4342	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑)
3	2	biimpri 230	. . . 4 ⊢ (∃𝑥 ∈ On 𝜑 → {𝑥 ∈ On ∣ 𝜑} ≠ ∅)
4		oninton 7774	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
5	1, 3, 4	sylancr 596	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
6		onminesb 7772	. . 3 ⊢ (∃𝑥 ∈ On 𝜑 → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
7		onss 7764	. . . . . . 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 3935	. . . . 5 ⊢ (∃𝑥 ∈ On 𝜑 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → 𝑦 ∈ On))
10		onminex.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	10	onnminsb 7778	. . . . 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 3152	. . 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 3316	. . . . 5 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (∀𝑦 ∈ 𝑧 ¬ 𝜓 ↔ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓))
16	14, 15	anbi12d 641	. . . 4 ⊢ (𝑧 = ∩ {𝑥 ∈ On ∣ 𝜑} → (([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓) ↔ ([∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑 ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} ¬ 𝜓)))
17	16	rspcev 3581	. . 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 847	. 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
19		nfv 1933	. . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓)
20		nfs1v 2189	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
21		nfv 1933	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 ¬ 𝜓
22	20, 21	nfan 1918	. . 3 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)
23		sbequ12 2285	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
24		raleq 3316	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
25	23, 24	anbi12d 641	. . 3 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓)))
26	19, 22, 25	cbvrexw 3304	. 2 ⊢ (∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓) ↔ ∃𝑧 ∈ On ([𝑧 / 𝑥]𝜑 ∧ ∀𝑦 ∈ 𝑧 ¬ 𝜓))
27	18, 26	sylibr 236	1 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  [wsb 2089   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  {crab 3413  [wsbc 3744   ⊆ wss 3904  ∅c0 4285  ∩ cint 4904  Oncon0 6342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-ord 6345  df-on 6346

This theorem is referenced by:  tz7.49  8411  omeulem1  8546  zorn2lem7  10456