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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 35365 | . . 3 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | |
2 | 1 | eleq2i 2819 | . 2 ⊢ (𝐴 ∈ Limits ↔ 𝐴 ∈ ((On ∩ Fix Bigcup ) ∖ {∅})) |
3 | eldif 3953 | . 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 6363 | . . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
6 | elin 3959 | . . . . 5 ⊢ (𝐴 ∈ (On ∩ Fix Bigcup ) ↔ (𝐴 ∈ On ∧ 𝐴 ∈ Fix Bigcup )) | |
7 | ellimits.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
8 | 7 | elon 6367 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ Ord 𝐴) |
9 | 7 | elfix 35408 | . . . . . . 7 ⊢ (𝐴 ∈ Fix Bigcup ↔ 𝐴 Bigcup 𝐴) |
10 | 7 | brbigcup 35403 | . . . . . . 7 ⊢ (𝐴 Bigcup 𝐴 ↔ ∪ 𝐴 = 𝐴) |
11 | eqcom 2733 | . . . . . . 7 ⊢ (∪ 𝐴 = 𝐴 ↔ 𝐴 = ∪ 𝐴) | |
12 | 9, 10, 11 | 3bitri 297 | . . . . . 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 275 | . . . 4 ⊢ (𝐴 ∈ (On ∩ Fix Bigcup ) ↔ (Ord 𝐴 ∧ 𝐴 = ∪ 𝐴)) |
15 | 7 | elsn 4638 | . . . . 5 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
16 | 15 | necon3bbii 2982 | . . . 4 ⊢ (¬ 𝐴 ∈ {∅} ↔ 𝐴 ≠ ∅) |
17 | 14, 16 | anbi12i 626 | . . 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 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 ∖ cdif 3940 ∩ cin 3942 ∅c0 4317 {csn 4623 ∪ cuni 4902 class class class wbr 5141 Ord word 6357 Oncon0 6358 Lim wlim 6359 Bigcup cbigcup 35339 Fix cfix 35340 Limits climits 35341 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-symdif 4237 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ord 6361 df-on 6362 df-lim 6363 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-1st 7974 df-2nd 7975 df-txp 35359 df-bigcup 35363 df-fix 35364 df-limits 35365 |
This theorem is referenced by: dfom5b 35417 dfrdg4 35456 |
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