Theorem onint 7732

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 7719	. . . 4 ⊢ Ord On
2		tz7.5 6335	. . . 4 ⊢ ((Ord On ∧ 𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅)
3	1, 2	mp3an1 1450	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅)
4		ssel 3925	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
5	4	imdistani 568	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (𝐴 ⊆ On ∧ 𝑥 ∈ On))
6		ssel 3925	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ On → (𝑧 ∈ 𝐴 → 𝑧 ∈ On))
7		ontri1 6348	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑥 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑥))
8		ssel 3925	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑧 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧))
9	7, 8	biimtrrdi 254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧 ∈ 𝑥 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧)))
10	9	ex 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (𝑧 ∈ On → (¬ 𝑧 ∈ 𝑥 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧))))
11	6, 10	sylan9 507	. . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧))
14	13	ralimdva 3146	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → (∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑥 → ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧))
15		disj 4401	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑥) = ∅ ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑥)
16		vex 3442	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
17	16	elint2 4906	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
18	14, 15, 17	3imtr4g 296	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → ((𝐴 ∩ 𝑥) = ∅ → 𝑦 ∈ ∩ 𝐴))
19	5, 18	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴)) → ((𝐴 ∩ 𝑥) = ∅ → 𝑦 ∈ ∩ 𝐴))
20	19	exp32 420	. . . . . . . . . . . . 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 418	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → (𝑦 ∈ 𝑥 → 𝑦 ∈ ∩ 𝐴))
23	22	ssrdv 3937	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 ⊆ ∩ 𝐴)
24		intss1 4915	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
25	24	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → ∩ 𝐴 ⊆ 𝑥)
26	23, 25	eqssd 3949	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 = ∩ 𝐴)
27	26	eleq1d 2818	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → (𝑥 ∈ 𝐴 ↔ ∩ 𝐴 ∈ 𝐴))
28	27	biimpd 229	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ 𝐴))
29	28	exp32 420	. . . . . 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 3133	. . 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 669	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058   ∩ cin 3898   ⊆ wss 3899  ∅c0 4284  ∩ cint 4899  Ord word 6313  Oncon0 6314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-br 5096  df-opab 5158  df-tr 5203  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318

This theorem is referenced by:  onint0  7733  onssmin  7734  onminesb  7735  onminsb  7736  oninton  7737  oneqmin  7742  oeeulem  8525  nnawordex  8561  unblem1  9186  unblem2  9187  tz9.12lem3  9692  scott0  9789  cardid2  9856  ackbij1lem18  10137  cardcf  10153  cff1  10159  cflim2  10164  cfss  10166  cofsmo  10170  fin23lem26  10226  pwfseqlem3  10561  gruina  10719  2ndcdisj  23381  sltval2  27605  nobdaymin  27726  lrrecfr  27896  rankeq1o  36226  dnnumch3  43154  oninfint  43343  inaex  44404