Theorem onint 7617

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 7604	. . . 4 ⊢ Ord On
2		tz7.5 6272	. . . 4 ⊢ ((Ord On ∧ 𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅)
3	1, 2	mp3an1 1446	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅)
4		ssel 3910	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
5	4	imdistani 568	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (𝐴 ⊆ On ∧ 𝑥 ∈ On))
6		ssel 3910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ On → (𝑧 ∈ 𝐴 → 𝑧 ∈ On))
7		ontri1 6285	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑥 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑥))
8		ssel 3910	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑧 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧))
9	7, 8	syl6bir 253	. . . . . . . . . . . . . . . . . . . . 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 3102	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ⊆ On ∧ 𝑥 ∈ On)) → (∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑥 → ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧))
15		disj 4378	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑥) = ∅ ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑥)
16		vex 3426	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
17	16	elint2 4883	. . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . 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 3923	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 ⊆ ∩ 𝐴)
24		intss1 4891	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
25	24	ad2antrl 724	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → ∩ 𝐴 ⊆ 𝑥)
26	23, 25	eqssd 3934	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 = ∩ 𝐴)
27	26	eleq1d 2823	. . . . . . . 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 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 3211	. . 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 665	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ∩ cint 4876  Ord word 6250  Oncon0 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255

This theorem is referenced by:  onint0  7618  onssmin  7619  onminesb  7620  onminsb  7621  oninton  7622  oneqmin  7627  oeeulem  8394  nnawordex  8430  unblem1  8996  unblem2  8997  tz9.12lem3  9478  scott0  9575  cardid2  9642  ackbij1lem18  9924  cardcf  9939  cff1  9945  cflim2  9950  cfss  9952  cofsmo  9956  fin23lem26  10012  pwfseqlem3  10347  gruina  10505  2ndcdisj  22515  sltval2  33786  nocvxmin  33900  lrrecfr  34027  rankeq1o  34400  dnnumch3  40788  inaex  41804