Theorem onint 7792

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 7778	. . . 4 ⊢ Ord On
2		tz7.5 6390	. . . 4 ⊢ ((Ord On ∧ 𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅)
3	1, 2	mp3an1 1444	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅)
4		ssel 3971	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
5	4	imdistani 567	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (𝐴 ⊆ On ∧ 𝑥 ∈ On))
6		ssel 3971	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ On → (𝑧 ∈ 𝐴 → 𝑧 ∈ On))
7		ontri1 6403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑥 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑥))
8		ssel 3971	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑧 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧))
9	7, 8	biimtrrdi 253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧 ∈ 𝑥 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧)))
10	9	ex 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (𝑧 ∈ On → (¬ 𝑧 ∈ 𝑥 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧))))
11	6, 10	sylan9 506	. . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧))
14	13	ralimdva 3157	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → (∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑥 → ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧))
15		disj 4448	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑥) = ∅ ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑥)
16		vex 3467	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
17	16	elint2 4956	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
18	14, 15, 17	3imtr4g 295	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → ((𝐴 ∩ 𝑥) = ∅ → 𝑦 ∈ ∩ 𝐴))
19	5, 18	sylan2 591	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴)) → ((𝐴 ∩ 𝑥) = ∅ → 𝑦 ∈ ∩ 𝐴))
20	19	exp32 419	. . . . . . . . . . . . 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 417	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → (𝑦 ∈ 𝑥 → 𝑦 ∈ ∩ 𝐴))
23	22	ssrdv 3983	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 ⊆ ∩ 𝐴)
24		intss1 4966	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
25	24	ad2antrl 726	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → ∩ 𝐴 ⊆ 𝑥)
26	23, 25	eqssd 3995	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 = ∩ 𝐴)
27	26	eleq1d 2810	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → (𝑥 ∈ 𝐴 ↔ ∩ 𝐴 ∈ 𝐴))
28	27	biimpd 228	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ 𝐴))
29	28	exp32 419	. . . . . 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 3143	. . 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 667	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ∀wral 3051  ∃wrex 3060   ∩ cin 3944   ⊆ wss 3945  ∅c0 4323  ∩ cint 4949  Ord word 6368  Oncon0 6369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6372  df-on 6373

This theorem is referenced by:  onint0  7793  onssmin  7794  onminesb  7795  onminsb  7796  oninton  7797  oneqmin  7802  oeeulem  8620  nnawordex  8656  unblem1  9318  unblem2  9319  tz9.12lem3  9812  scott0  9909  cardid2  9976  ackbij1lem18  10260  cardcf  10275  cff1  10281  cflim2  10286  cfss  10288  cofsmo  10292  fin23lem26  10348  pwfseqlem3  10683  gruina  10841  2ndcdisj  23390  sltval2  27619  nocvxmin  27741  lrrecfr  27890  rankeq1o  35837  dnnumch3  42536  oninfint  42729  inaex  43799