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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 9158 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
3 | 2 | ensymd 9007 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≈ 𝐴) |
4 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
5 | en2sn 9047 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → {∅} ≈ {𝐴}) | |
6 | 4, 5 | mpan 687 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {∅} ≈ {𝐴}) |
7 | disjdifr 4472 | . . . 4 ⊢ ((𝐴 ∖ {∅}) ∩ {∅}) = ∅ | |
8 | limord 6424 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
9 | 1, 8 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
10 | ordirr 6382 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ ¬ 𝐴 ∈ 𝐴 |
12 | disjsn 4715 | . . . . 5 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
13 | 11, 12 | mpbir 230 | . . . 4 ⊢ (𝐴 ∩ {𝐴}) = ∅ |
14 | unen 9052 | . . . 4 ⊢ ((((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) ∧ (((𝐴 ∖ {∅}) ∩ {∅}) = ∅ ∧ (𝐴 ∩ {𝐴}) = ∅)) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) | |
15 | 7, 13, 14 | mpanr12 702 | . . 3 ⊢ (((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
16 | 3, 6, 15 | syl2anc 583 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
17 | 0ellim 6427 | . . . . . 6 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) | |
18 | 1, 17 | ax-mp 5 | . . . . 5 ⊢ ∅ ∈ 𝐴 |
19 | 4 | snss 4789 | . . . . 5 ⊢ (∅ ∈ 𝐴 ↔ {∅} ⊆ 𝐴) |
20 | 18, 19 | mpbi 229 | . . . 4 ⊢ {∅} ⊆ 𝐴 |
21 | undif 4481 | . . . 4 ⊢ ({∅} ⊆ 𝐴 ↔ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴) | |
22 | 20, 21 | mpbi 229 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴 |
23 | uncom 4153 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = ((𝐴 ∖ {∅}) ∪ {∅}) | |
24 | 22, 23 | eqtr3i 2761 | . 2 ⊢ 𝐴 = ((𝐴 ∖ {∅}) ∪ {∅}) |
25 | df-suc 6370 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
26 | 16, 24, 25 | 3brtr4g 5182 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {csn 4628 class class class wbr 5148 Ord word 6363 Lim wlim 6365 suc csuc 6366 ≈ cen 8942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8709 df-en 8946 df-dom 8947 |
This theorem is referenced by: limensuc 9160 infensuc 9161 omensuc 9657 |
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