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Mirrors > Home > MPE Home > Th. List > iinon | Structured version Visualization version GIF version |
Description: The nonempty indexed intersection of a class of ordinal numbers 𝐵(𝑥) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Ref | Expression |
---|---|
iinon | ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiin3g 5829 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
2 | 1 | adantr 481 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
3 | eqid 2818 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | rnmptss 6878 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On) |
5 | dm0rn0 5788 | . . . . . 6 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅) | |
6 | dmmptg 6089 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) | |
7 | 6 | eqeq1d 2820 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅)) |
8 | 5, 7 | syl5bbr 286 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅)) |
9 | 8 | necon3bid 3057 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅)) |
10 | 9 | biimpar 478 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) |
11 | oninton 7504 | . . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On) | |
12 | 4, 10, 11 | syl2an2r 681 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On) |
13 | 2, 12 | eqeltrd 2910 | 1 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ⊆ wss 3933 ∅c0 4288 ∩ cint 4867 ∩ ciin 4911 ↦ cmpt 5137 dom cdm 5548 ran crn 5549 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 |
This theorem is referenced by: (None) |
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