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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 5335 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ∈ V) | |
2 | limenpsi.1 | . . . . . . 7 ⊢ Lim 𝐴 | |
3 | limsuc 7870 | . . . . . . 7 ⊢ (Lim 𝐴 → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴) |
5 | 4 | biimpi 216 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) |
6 | nsuceq0 6469 | . . . . 5 ⊢ suc 𝑥 ≠ ∅ | |
7 | eldifsn 4791 | . . . . 5 ⊢ (suc 𝑥 ∈ (𝐴 ∖ {∅}) ↔ (suc 𝑥 ∈ 𝐴 ∧ suc 𝑥 ≠ ∅)) | |
8 | 5, 6, 7 | sylanblrc 590 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ (𝐴 ∖ {∅})) |
9 | limord 6446 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
10 | 2, 9 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
11 | ordelon 6410 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
12 | 10, 11 | mpan 690 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On) |
13 | ordelon 6410 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On) | |
14 | 10, 13 | mpan 690 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ On) |
15 | suc11 6493 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦)) | |
16 | 12, 14, 15 | syl2an 596 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦)) |
17 | 8, 16 | dom3 9035 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∖ {∅}) ∈ V) → 𝐴 ≼ (𝐴 ∖ {∅})) |
18 | 1, 17 | mpdan 687 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝐴 ∖ {∅})) |
19 | difss 4146 | . . 3 ⊢ (𝐴 ∖ {∅}) ⊆ 𝐴 | |
20 | ssdomg 9039 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ⊆ 𝐴 → (𝐴 ∖ {∅}) ≼ 𝐴)) | |
21 | 19, 20 | mpi 20 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≼ 𝐴) |
22 | sbth 9132 | . 2 ⊢ ((𝐴 ≼ (𝐴 ∖ {∅}) ∧ (𝐴 ∖ {∅}) ≼ 𝐴) → 𝐴 ≈ (𝐴 ∖ {∅})) | |
23 | 18, 21, 22 | syl2anc 584 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 {csn 4631 class class class wbr 5148 Ord word 6385 Oncon0 6386 Lim wlim 6387 suc csuc 6388 ≈ cen 8981 ≼ cdom 8982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-en 8985 df-dom 8986 |
This theorem is referenced by: limensuci 9192 omenps 9693 infdifsn 9695 ominf4 10350 |
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