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Mirrors > Home > MPE Home > Th. List > limenpsi | Structured version Visualization version GIF version |
Description: A limit ordinal is equinumerous to a proper subset of itself. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
limenpsi.1 | ⊢ Lim 𝐴 |
Ref | Expression |
---|---|
limenpsi | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexg 4808 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ∈ V) | |
2 | limenpsi.1 | . . . . . . . 8 ⊢ Lim 𝐴 | |
3 | limsuc 7049 | . . . . . . . 8 ⊢ (Lim 𝐴 → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴) |
5 | 4 | biimpi 206 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) |
6 | nsuceq0 5805 | . . . . . 6 ⊢ suc 𝑥 ≠ ∅ | |
7 | 5, 6 | jctir 561 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (suc 𝑥 ∈ 𝐴 ∧ suc 𝑥 ≠ ∅)) |
8 | eldifsn 4317 | . . . . 5 ⊢ (suc 𝑥 ∈ (𝐴 ∖ {∅}) ↔ (suc 𝑥 ∈ 𝐴 ∧ suc 𝑥 ≠ ∅)) | |
9 | 7, 8 | sylibr 224 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ (𝐴 ∖ {∅})) |
10 | limord 5784 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
11 | 2, 10 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
12 | ordelon 5747 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
13 | 11, 12 | mpan 706 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On) |
14 | ordelon 5747 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On) | |
15 | 11, 14 | mpan 706 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ On) |
16 | suc11 5831 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦)) | |
17 | 13, 15, 16 | syl2an 494 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦)) |
18 | 9, 17 | dom3 7999 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∖ {∅}) ∈ V) → 𝐴 ≼ (𝐴 ∖ {∅})) |
19 | 1, 18 | mpdan 702 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝐴 ∖ {∅})) |
20 | difss 3737 | . . 3 ⊢ (𝐴 ∖ {∅}) ⊆ 𝐴 | |
21 | ssdomg 8001 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ⊆ 𝐴 → (𝐴 ∖ {∅}) ≼ 𝐴)) | |
22 | 20, 21 | mpi 20 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≼ 𝐴) |
23 | sbth 8080 | . 2 ⊢ ((𝐴 ≼ (𝐴 ∖ {∅}) ∧ (𝐴 ∖ {∅}) ≼ 𝐴) → 𝐴 ≈ (𝐴 ∖ {∅})) | |
24 | 19, 22, 23 | syl2anc 693 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 {csn 4177 class class class wbr 4653 Ord word 5722 Oncon0 5723 Lim wlim 5724 suc csuc 5725 ≈ cen 7952 ≼ cdom 7953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-en 7956 df-dom 7957 |
This theorem is referenced by: limensuci 8136 omenps 8552 infdifsn 8554 ominf4 9134 |
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