Theorem onint 7769

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 7756	. . . 4 ⊢ Ord On
2		tz7.5 6356	. . . 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 3943	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
5	4	imdistani 568	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (𝐴 ⊆ On ∧ 𝑥 ∈ On))
6		ssel 3943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ On → (𝑧 ∈ 𝐴 → 𝑧 ∈ On))
7		ontri1 6369	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑥 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑥))
8		ssel 3943	. . . . . . . . . . . . . . . . . . . . . 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 4416	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑥) = ∅ ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑥)
16		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
17	16	elint2 4920	. . . . . . . . . . . . . . . 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 3955	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 ⊆ ∩ 𝐴)
24		intss1 4930	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
25	24	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → ∩ 𝐴 ⊆ 𝑥)
26	23, 25	eqssd 3967	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ (𝐴 ∩ 𝑥) = ∅)) → 𝑥 = ∩ 𝐴)
27	26	eleq1d 2814	. . . . . . . 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 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  ∩ cint 4913  Ord word 6334  Oncon0 6335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339

This theorem is referenced by:  onint0  7770  onssmin  7771  onminesb  7772  onminsb  7773  oninton  7774  oneqmin  7779  oeeulem  8568  nnawordex  8604  unblem1  9246  unblem2  9247  tz9.12lem3  9749  scott0  9846  cardid2  9913  ackbij1lem18  10196  cardcf  10212  cff1  10218  cflim2  10223  cfss  10225  cofsmo  10229  fin23lem26  10285  pwfseqlem3  10620  gruina  10778  2ndcdisj  23350  sltval2  27575  nocvxmin  27697  lrrecfr  27857  rankeq1o  36166  dnnumch3  43043  oninfint  43232  inaex  44293