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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 33325 | . . 3 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | |
2 | 1 | eleq2i 2907 | . 2 ⊢ (𝐴 ∈ Limits ↔ 𝐴 ∈ ((On ∩ Fix Bigcup ) ∖ {∅})) |
3 | eldif 3949 | . 2 ⊢ (𝐴 ∈ ((On ∩ Fix Bigcup ) ∖ {∅}) ↔ (𝐴 ∈ (On ∩ Fix Bigcup ) ∧ ¬ 𝐴 ∈ {∅})) | |
4 | 3anan32 1093 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) ∧ 𝐴 ≠ ∅)) | |
5 | df-lim 6199 | . . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
6 | elin 4172 | . . . . 5 ⊢ (𝐴 ∈ (On ∩ Fix Bigcup ) ↔ (𝐴 ∈ On ∧ 𝐴 ∈ Fix Bigcup )) | |
7 | ellimits.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
8 | 7 | elon 6203 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ Ord 𝐴) |
9 | 7 | elfix 33368 | . . . . . . 7 ⊢ (𝐴 ∈ Fix Bigcup ↔ 𝐴 Bigcup 𝐴) |
10 | 7 | brbigcup 33363 | . . . . . . 7 ⊢ (𝐴 Bigcup 𝐴 ↔ ∪ 𝐴 = 𝐴) |
11 | eqcom 2831 | . . . . . . 7 ⊢ (∪ 𝐴 = 𝐴 ↔ 𝐴 = ∪ 𝐴) | |
12 | 9, 10, 11 | 3bitri 299 | . . . . . 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 277 | . . . 4 ⊢ (𝐴 ∈ (On ∩ Fix Bigcup ) ↔ (Ord 𝐴 ∧ 𝐴 = ∪ 𝐴)) |
15 | 7 | elsn 4585 | . . . . 5 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
16 | 15 | necon3bbii 3066 | . . . 4 ⊢ (¬ 𝐴 ∈ {∅} ↔ 𝐴 ≠ ∅) |
17 | 14, 16 | anbi12i 628 | . . 3 ⊢ ((𝐴 ∈ (On ∩ Fix Bigcup ) ∧ ¬ 𝐴 ∈ {∅}) ↔ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) ∧ 𝐴 ≠ ∅)) |
18 | 4, 5, 17 | 3bitr4ri 306 | . 2 ⊢ ((𝐴 ∈ (On ∩ Fix Bigcup ) ∧ ¬ 𝐴 ∈ {∅}) ↔ Lim 𝐴) |
19 | 2, 3, 18 | 3bitri 299 | 1 ⊢ (𝐴 ∈ Limits ↔ Lim 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∖ cdif 3936 ∩ cin 3938 ∅c0 4294 {csn 4570 ∪ cuni 4841 class class class wbr 5069 Ord word 6193 Oncon0 6194 Lim wlim 6195 Bigcup cbigcup 33299 Fix cfix 33300 Limits climits 33301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-symdif 4222 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ord 6197 df-on 6198 df-lim 6199 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-1st 7692 df-2nd 7693 df-txp 33319 df-bigcup 33323 df-fix 33324 df-limits 33325 |
This theorem is referenced by: dfom5b 33377 dfrdg4 33416 |
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