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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 35694 | . . 3 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | |
2 | 1 | eleq2i 2818 | . 2 ⊢ (𝐴 ∈ Limits ↔ 𝐴 ∈ ((On ∩ Fix Bigcup ) ∖ {∅})) |
3 | eldif 3956 | . 2 ⊢ (𝐴 ∈ ((On ∩ Fix Bigcup ) ∖ {∅}) ↔ (𝐴 ∈ (On ∩ Fix Bigcup ) ∧ ¬ 𝐴 ∈ {∅})) | |
4 | 3anan32 1094 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) ∧ 𝐴 ≠ ∅)) | |
5 | df-lim 6370 | . . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
6 | elin 3962 | . . . . 5 ⊢ (𝐴 ∈ (On ∩ Fix Bigcup ) ↔ (𝐴 ∈ On ∧ 𝐴 ∈ Fix Bigcup )) | |
7 | ellimits.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
8 | 7 | elon 6374 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ Ord 𝐴) |
9 | 7 | elfix 35737 | . . . . . . 7 ⊢ (𝐴 ∈ Fix Bigcup ↔ 𝐴 Bigcup 𝐴) |
10 | 7 | brbigcup 35732 | . . . . . . 7 ⊢ (𝐴 Bigcup 𝐴 ↔ ∪ 𝐴 = 𝐴) |
11 | eqcom 2733 | . . . . . . 7 ⊢ (∪ 𝐴 = 𝐴 ↔ 𝐴 = ∪ 𝐴) | |
12 | 9, 10, 11 | 3bitri 296 | . . . . . 6 ⊢ (𝐴 ∈ Fix Bigcup ↔ 𝐴 = ∪ 𝐴) |
13 | 8, 12 | anbi12i 626 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ Fix Bigcup ) ↔ (Ord 𝐴 ∧ 𝐴 = ∪ 𝐴)) |
14 | 6, 13 | bitri 274 | . . . 4 ⊢ (𝐴 ∈ (On ∩ Fix Bigcup ) ↔ (Ord 𝐴 ∧ 𝐴 = ∪ 𝐴)) |
15 | 7 | elsn 4638 | . . . . 5 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
16 | 15 | necon3bbii 2978 | . . . 4 ⊢ (¬ 𝐴 ∈ {∅} ↔ 𝐴 ≠ ∅) |
17 | 14, 16 | anbi12i 626 | . . 3 ⊢ ((𝐴 ∈ (On ∩ Fix Bigcup ) ∧ ¬ 𝐴 ∈ {∅}) ↔ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) ∧ 𝐴 ≠ ∅)) |
18 | 4, 5, 17 | 3bitr4ri 303 | . 2 ⊢ ((𝐴 ∈ (On ∩ Fix Bigcup ) ∧ ¬ 𝐴 ∈ {∅}) ↔ Lim 𝐴) |
19 | 2, 3, 18 | 3bitri 296 | 1 ⊢ (𝐴 ∈ Limits ↔ Lim 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ∖ cdif 3943 ∩ cin 3945 ∅c0 4322 {csn 4623 ∪ cuni 4905 class class class wbr 5143 Ord word 6364 Oncon0 6365 Lim wlim 6366 Bigcup cbigcup 35668 Fix cfix 35669 Limits climits 35670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-symdif 4241 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ord 6368 df-on 6369 df-lim 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7992 df-2nd 7993 df-txp 35688 df-bigcup 35692 df-fix 35693 df-limits 35694 |
This theorem is referenced by: dfom5b 35746 dfrdg4 35785 |
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