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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 35497 | . . 3 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | |
2 | 1 | eleq2i 2821 | . 2 ⊢ (𝐴 ∈ Limits ↔ 𝐴 ∈ ((On ∩ Fix Bigcup ) ∖ {∅})) |
3 | eldif 3959 | . 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 6379 | . . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
6 | elin 3965 | . . . . 5 ⊢ (𝐴 ∈ (On ∩ Fix Bigcup ) ↔ (𝐴 ∈ On ∧ 𝐴 ∈ Fix Bigcup )) | |
7 | ellimits.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
8 | 7 | elon 6383 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ Ord 𝐴) |
9 | 7 | elfix 35540 | . . . . . . 7 ⊢ (𝐴 ∈ Fix Bigcup ↔ 𝐴 Bigcup 𝐴) |
10 | 7 | brbigcup 35535 | . . . . . . 7 ⊢ (𝐴 Bigcup 𝐴 ↔ ∪ 𝐴 = 𝐴) |
11 | eqcom 2735 | . . . . . . 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 4647 | . . . . 5 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
16 | 15 | necon3bbii 2985 | . . . 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 1533 ∈ wcel 2098 ≠ wne 2937 Vcvv 3473 ∖ cdif 3946 ∩ cin 3948 ∅c0 4326 {csn 4632 ∪ cuni 4912 class class class wbr 5152 Ord word 6373 Oncon0 6374 Lim wlim 6375 Bigcup cbigcup 35471 Fix cfix 35472 Limits climits 35473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-symdif 4245 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ord 6377 df-on 6378 df-lim 6379 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-fv 6561 df-1st 8001 df-2nd 8002 df-txp 35491 df-bigcup 35495 df-fix 35496 df-limits 35497 |
This theorem is referenced by: dfom5b 35549 dfrdg4 35588 |
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