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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 8706 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
3 | 2 | ensymd 8571 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≈ 𝐴) |
4 | 0ex 5170 | . . . 4 ⊢ ∅ ∈ V | |
5 | en2sn 8605 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → {∅} ≈ {𝐴}) | |
6 | 4, 5 | mpan 690 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {∅} ≈ {𝐴}) |
7 | incom 4102 | . . . . 5 ⊢ ((𝐴 ∖ {∅}) ∩ {∅}) = ({∅} ∩ (𝐴 ∖ {∅})) | |
8 | disjdif 4361 | . . . . 5 ⊢ ({∅} ∩ (𝐴 ∖ {∅})) = ∅ | |
9 | 7, 8 | eqtri 2782 | . . . 4 ⊢ ((𝐴 ∖ {∅}) ∩ {∅}) = ∅ |
10 | limord 6221 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
11 | 1, 10 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
12 | ordirr 6180 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ ¬ 𝐴 ∈ 𝐴 |
14 | disjsn 4597 | . . . . 5 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
15 | 13, 14 | mpbir 234 | . . . 4 ⊢ (𝐴 ∩ {𝐴}) = ∅ |
16 | unen 8609 | . . . 4 ⊢ ((((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) ∧ (((𝐴 ∖ {∅}) ∩ {∅}) = ∅ ∧ (𝐴 ∩ {𝐴}) = ∅)) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) | |
17 | 9, 15, 16 | mpanr12 705 | . . 3 ⊢ (((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
18 | 3, 6, 17 | syl2anc 588 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
19 | 0ellim 6224 | . . . . . 6 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) | |
20 | 1, 19 | ax-mp 5 | . . . . 5 ⊢ ∅ ∈ 𝐴 |
21 | 4 | snss 4669 | . . . . 5 ⊢ (∅ ∈ 𝐴 ↔ {∅} ⊆ 𝐴) |
22 | 20, 21 | mpbi 233 | . . . 4 ⊢ {∅} ⊆ 𝐴 |
23 | undif 4371 | . . . 4 ⊢ ({∅} ⊆ 𝐴 ↔ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴) | |
24 | 22, 23 | mpbi 233 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴 |
25 | uncom 4054 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = ((𝐴 ∖ {∅}) ∪ {∅}) | |
26 | 24, 25 | eqtr3i 2784 | . 2 ⊢ 𝐴 = ((𝐴 ∖ {∅}) ∪ {∅}) |
27 | df-suc 6168 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
28 | 18, 26, 27 | 3brtr4g 5059 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ∖ cdif 3851 ∪ cun 3852 ∩ cin 3853 ⊆ wss 3854 ∅c0 4221 {csn 4515 class class class wbr 5025 Ord word 6161 Lim wlim 6163 suc csuc 6164 ≈ cen 8517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-er 8292 df-en 8521 df-dom 8522 |
This theorem is referenced by: limensuc 8708 infensuc 8709 omensuc 9137 |
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