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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 8680 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
3 | 2 | ensymd 8548 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≈ 𝐴) |
4 | 0ex 5202 | . . . 4 ⊢ ∅ ∈ V | |
5 | en2sn 8581 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → {∅} ≈ {𝐴}) | |
6 | 4, 5 | mpan 686 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {∅} ≈ {𝐴}) |
7 | incom 4175 | . . . . 5 ⊢ ((𝐴 ∖ {∅}) ∩ {∅}) = ({∅} ∩ (𝐴 ∖ {∅})) | |
8 | disjdif 4417 | . . . . 5 ⊢ ({∅} ∩ (𝐴 ∖ {∅})) = ∅ | |
9 | 7, 8 | eqtri 2841 | . . . 4 ⊢ ((𝐴 ∖ {∅}) ∩ {∅}) = ∅ |
10 | limord 6243 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
11 | 1, 10 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
12 | ordirr 6202 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ ¬ 𝐴 ∈ 𝐴 |
14 | disjsn 4639 | . . . . 5 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
15 | 13, 14 | mpbir 232 | . . . 4 ⊢ (𝐴 ∩ {𝐴}) = ∅ |
16 | unen 8584 | . . . 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 6246 | . . . . . 6 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) | |
20 | 1, 19 | ax-mp 5 | . . . . 5 ⊢ ∅ ∈ 𝐴 |
21 | 4 | snss 4710 | . . . . 5 ⊢ (∅ ∈ 𝐴 ↔ {∅} ⊆ 𝐴) |
22 | 20, 21 | mpbi 231 | . . . 4 ⊢ {∅} ⊆ 𝐴 |
23 | undif 4426 | . . . 4 ⊢ ({∅} ⊆ 𝐴 ↔ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴) | |
24 | 22, 23 | mpbi 231 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴 |
25 | uncom 4126 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = ((𝐴 ∖ {∅}) ∪ {∅}) | |
26 | 24, 25 | eqtr3i 2843 | . 2 ⊢ 𝐴 = ((𝐴 ∖ {∅}) ∪ {∅}) |
27 | df-suc 6190 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
28 | 18, 26, 27 | 3brtr4g 5091 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 {csn 4557 class class class wbr 5057 Ord word 6183 Lim wlim 6185 suc csuc 6186 ≈ cen 8494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 |
This theorem is referenced by: limensuc 8682 infensuc 8683 omensuc 9107 |
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