![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ellimits | Structured version Visualization version GIF version |
Description: Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
ellimits.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ellimits | ⊢ (𝐴 ∈ Limits ↔ Lim 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-limits 35841 | . . 3 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ Limits ↔ 𝐴 ∈ ((On ∩ Fix Bigcup ) ∖ {∅})) |
3 | eldif 3972 | . 2 ⊢ (𝐴 ∈ ((On ∩ Fix Bigcup ) ∖ {∅}) ↔ (𝐴 ∈ (On ∩ Fix Bigcup ) ∧ ¬ 𝐴 ∈ {∅})) | |
4 | 3anan32 1096 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) ∧ 𝐴 ≠ ∅)) | |
5 | df-lim 6390 | . . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
6 | elin 3978 | . . . . 5 ⊢ (𝐴 ∈ (On ∩ Fix Bigcup ) ↔ (𝐴 ∈ On ∧ 𝐴 ∈ Fix Bigcup )) | |
7 | ellimits.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
8 | 7 | elon 6394 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ Ord 𝐴) |
9 | 7 | elfix 35884 | . . . . . . 7 ⊢ (𝐴 ∈ Fix Bigcup ↔ 𝐴 Bigcup 𝐴) |
10 | 7 | brbigcup 35879 | . . . . . . 7 ⊢ (𝐴 Bigcup 𝐴 ↔ ∪ 𝐴 = 𝐴) |
11 | eqcom 2741 | . . . . . . 7 ⊢ (∪ 𝐴 = 𝐴 ↔ 𝐴 = ∪ 𝐴) | |
12 | 9, 10, 11 | 3bitri 297 | . . . . . 6 ⊢ (𝐴 ∈ Fix Bigcup ↔ 𝐴 = ∪ 𝐴) |
13 | 8, 12 | anbi12i 628 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ Fix Bigcup ) ↔ (Ord 𝐴 ∧ 𝐴 = ∪ 𝐴)) |
14 | 6, 13 | bitri 275 | . . . 4 ⊢ (𝐴 ∈ (On ∩ Fix Bigcup ) ↔ (Ord 𝐴 ∧ 𝐴 = ∪ 𝐴)) |
15 | 7 | elsn 4645 | . . . . 5 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
16 | 15 | necon3bbii 2985 | . . . 4 ⊢ (¬ 𝐴 ∈ {∅} ↔ 𝐴 ≠ ∅) |
17 | 14, 16 | anbi12i 628 | . . 3 ⊢ ((𝐴 ∈ (On ∩ Fix Bigcup ) ∧ ¬ 𝐴 ∈ {∅}) ↔ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) ∧ 𝐴 ≠ ∅)) |
18 | 4, 5, 17 | 3bitr4ri 304 | . 2 ⊢ ((𝐴 ∈ (On ∩ Fix Bigcup ) ∧ ¬ 𝐴 ∈ {∅}) ↔ Lim 𝐴) |
19 | 2, 3, 18 | 3bitri 297 | 1 ⊢ (𝐴 ∈ Limits ↔ Lim 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∖ cdif 3959 ∩ cin 3961 ∅c0 4338 {csn 4630 ∪ cuni 4911 class class class wbr 5147 Ord word 6384 Oncon0 6385 Lim wlim 6386 Bigcup cbigcup 35815 Fix cfix 35816 Limits climits 35817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-symdif 4258 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ord 6388 df-on 6389 df-lim 6390 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fo 6568 df-fv 6570 df-1st 8012 df-2nd 8013 df-txp 35835 df-bigcup 35839 df-fix 35840 df-limits 35841 |
This theorem is referenced by: dfom5b 35893 dfrdg4 35932 |
Copyright terms: Public domain | W3C validator |