Theorem onint 7490

Description: The intersection (infimum) of a nonempty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.)

Assertion

Ref	Expression
onint	⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴)

Proof of Theorem onint

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordon 7478	. . . 4 ⊢ Ord On
2		tz7.5 6180	. . . 4 ⊢ ((Ord On ∧ 𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅)
3	1, 2	mp3an1 1445	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅)
4		ssel 3908	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
5	4	imdistani 572	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (𝐴 ⊆ On ∧ 𝑥 ∈ On))
6		ssel 3908	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ On → (𝑧 ∈ 𝐴 → 𝑧 ∈ On))
7		ontri1 6193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑥 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑥))
8		ssel 3908	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑧 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧))
9	7, 8	syl6bir 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧 ∈ 𝑥 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧)))
10	9	ex 416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (𝑧 ∈ On → (¬ 𝑧 ∈ 𝑥 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧))))
11	6, 10	sylan9 511	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ On) → (𝑧 ∈ 𝐴 → (¬ 𝑧 ∈ 𝑥 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧))))
12	11	com4r 94	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝑥 → ((𝐴 ⊆ On ∧ 𝑥 ∈ On) → (𝑧 ∈ 𝐴 → (¬ 𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧))))
13	12	imp31 421	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧))
14	13	ralimdva 3144	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → (∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑥 → ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧))
15		disj 4355	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑥) = ∅ ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑥)
16		vex 3444	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
17	16	elint2 4845	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
18	14, 15, 17	3imtr4g 299	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → ((𝐴 ∩ 𝑥) = ∅ → 𝑦 ∈ ∩ 𝐴))
19	5, 18	sylan2 595	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴)) → ((𝐴 ∩ 𝑥) = ∅ → 𝑦 ∈ ∩ 𝐴))
20	19	exp32 424	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑥 → (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → ((𝐴 ∩ 𝑥) = ∅ → 𝑦 ∈ ∩ 𝐴))))
21	20	com4l 92	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → ((𝐴 ∩ 𝑥) = ∅ → (𝑦 ∈ 𝑥 → 𝑦 ∈ ∩ 𝐴))))
22	21	imp32 422	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → (𝑦 ∈ 𝑥 → 𝑦 ∈ ∩ 𝐴))
23	22	ssrdv 3921	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 ⊆ ∩ 𝐴)
24		intss1 4853	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
25	24	ad2antrl 727	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → ∩ 𝐴 ⊆ 𝑥)
26	23, 25	eqssd 3932	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 = ∩ 𝐴)
27	26	eleq1d 2874	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → (𝑥 ∈ 𝐴 ↔ ∩ 𝐴 ∈ 𝐴))
28	27	biimpd 232	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ 𝐴))
29	28	exp32 424	. . . . . 6 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → ((𝐴 ∩ 𝑥) = ∅ → (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ 𝐴))))
30	29	com34 91	. . . . 5 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝐴 ∩ 𝑥) = ∅ → ∩ 𝐴 ∈ 𝐴))))
31	30	pm2.43d 53	. . . 4 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → ((𝐴 ∩ 𝑥) = ∅ → ∩ 𝐴 ∈ 𝐴)))
32	31	rexlimdv 3242	. . 3 ⊢ (𝐴 ⊆ On → (∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅ → ∩ 𝐴 ∈ 𝐴))
33	3, 32	syl5 34	. 2 ⊢ (𝐴 ⊆ On → ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴))
34	33	anabsi5 668	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ∩ cint 4838  Ord word 6158  Oncon0 6159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163

This theorem is referenced by:  onint0  7491  onssmin  7492  onminesb  7493  onminsb  7494  oninton  7495  oneqmin  7500  oeeulem  8210  nnawordex  8246  unblem1  8754  unblem2  8755  tz9.12lem3  9202  scott0  9299  cardid2  9366  ackbij1lem18  9648  cardcf  9663  cff1  9669  cflim2  9674  cfss  9676  cofsmo  9680  fin23lem26  9736  pwfseqlem3  10071  gruina  10229  2ndcdisj  22061  sltval2  33276  nocvxmin  33361  rankeq1o  33745  dnnumch3  39991  inaex  41005