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Mirrors > Home > MPE Home > Th. List > limensuci | Structured version Visualization version GIF version |
Description: A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.) |
Ref | Expression |
---|---|
limensuci.1 | ⊢ Lim 𝐴 |
Ref | Expression |
---|---|
limensuci | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limensuci.1 | . . . . 5 ⊢ Lim 𝐴 | |
2 | 1 | limenpsi 8546 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
3 | 2 | ensymd 8415 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≈ 𝐴) |
4 | 0ex 5109 | . . . 4 ⊢ ∅ ∈ V | |
5 | en2sn 8448 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → {∅} ≈ {𝐴}) | |
6 | 4, 5 | mpan 686 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {∅} ≈ {𝐴}) |
7 | incom 4105 | . . . . 5 ⊢ ((𝐴 ∖ {∅}) ∩ {∅}) = ({∅} ∩ (𝐴 ∖ {∅})) | |
8 | disjdif 4341 | . . . . 5 ⊢ ({∅} ∩ (𝐴 ∖ {∅})) = ∅ | |
9 | 7, 8 | eqtri 2821 | . . . 4 ⊢ ((𝐴 ∖ {∅}) ∩ {∅}) = ∅ |
10 | limord 6132 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
11 | 1, 10 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
12 | ordirr 6091 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ ¬ 𝐴 ∈ 𝐴 |
14 | disjsn 4560 | . . . . 5 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
15 | 13, 14 | mpbir 232 | . . . 4 ⊢ (𝐴 ∩ {𝐴}) = ∅ |
16 | unen 8451 | . . . 4 ⊢ ((((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) ∧ (((𝐴 ∖ {∅}) ∩ {∅}) = ∅ ∧ (𝐴 ∩ {𝐴}) = ∅)) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) | |
17 | 9, 15, 16 | mpanr12 701 | . . 3 ⊢ (((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
18 | 3, 6, 17 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
19 | 0ellim 6135 | . . . . . 6 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) | |
20 | 1, 19 | ax-mp 5 | . . . . 5 ⊢ ∅ ∈ 𝐴 |
21 | 4 | snss 4631 | . . . . 5 ⊢ (∅ ∈ 𝐴 ↔ {∅} ⊆ 𝐴) |
22 | 20, 21 | mpbi 231 | . . . 4 ⊢ {∅} ⊆ 𝐴 |
23 | undif 4350 | . . . 4 ⊢ ({∅} ⊆ 𝐴 ↔ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴) | |
24 | 22, 23 | mpbi 231 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴 |
25 | uncom 4056 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = ((𝐴 ∖ {∅}) ∪ {∅}) | |
26 | 24, 25 | eqtr3i 2823 | . 2 ⊢ 𝐴 = ((𝐴 ∖ {∅}) ∪ {∅}) |
27 | df-suc 6079 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
28 | 18, 26, 27 | 3brtr4g 5002 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 Vcvv 3440 ∖ cdif 3862 ∪ cun 3863 ∩ cin 3864 ⊆ wss 3865 ∅c0 4217 {csn 4478 class class class wbr 4968 Ord word 6072 Lim wlim 6074 suc csuc 6075 ≈ cen 8361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-1o 7960 df-er 8146 df-en 8365 df-dom 8366 |
This theorem is referenced by: limensuc 8548 infensuc 8549 omensuc 8972 |
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