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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 9191 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
3 | 2 | ensymd 9044 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≈ 𝐴) |
4 | 0ex 5313 | . . . 4 ⊢ ∅ ∈ V | |
5 | en2sn 9080 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → {∅} ≈ {𝐴}) | |
6 | 4, 5 | mpan 690 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {∅} ≈ {𝐴}) |
7 | disjdifr 4479 | . . . 4 ⊢ ((𝐴 ∖ {∅}) ∩ {∅}) = ∅ | |
8 | limord 6446 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
9 | 1, 8 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
10 | ordirr 6404 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ ¬ 𝐴 ∈ 𝐴 |
12 | disjsn 4716 | . . . . 5 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
13 | 11, 12 | mpbir 231 | . . . 4 ⊢ (𝐴 ∩ {𝐴}) = ∅ |
14 | unen 9085 | . . . 4 ⊢ ((((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) ∧ (((𝐴 ∖ {∅}) ∩ {∅}) = ∅ ∧ (𝐴 ∩ {𝐴}) = ∅)) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) | |
15 | 7, 13, 14 | mpanr12 705 | . . 3 ⊢ (((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
16 | 3, 6, 15 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
17 | 0ellim 6449 | . . . . . 6 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) | |
18 | 1, 17 | ax-mp 5 | . . . . 5 ⊢ ∅ ∈ 𝐴 |
19 | 4 | snss 4790 | . . . . 5 ⊢ (∅ ∈ 𝐴 ↔ {∅} ⊆ 𝐴) |
20 | 18, 19 | mpbi 230 | . . . 4 ⊢ {∅} ⊆ 𝐴 |
21 | undif 4488 | . . . 4 ⊢ ({∅} ⊆ 𝐴 ↔ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴) | |
22 | 20, 21 | mpbi 230 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴 |
23 | uncom 4168 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = ((𝐴 ∖ {∅}) ∪ {∅}) | |
24 | 22, 23 | eqtr3i 2765 | . 2 ⊢ 𝐴 = ((𝐴 ∖ {∅}) ∪ {∅}) |
25 | df-suc 6392 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {csn 4631 class class class wbr 5148 Ord word 6385 Lim wlim 6387 suc csuc 6388 ≈ cen 8981 |
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-er 8744 df-en 8985 df-dom 8986 |
This theorem is referenced by: limensuc 9193 infensuc 9194 omensuc 9694 |
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