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Mirrors > Home > MPE Home > Th. List > suceloni | Structured version Visualization version GIF version |
Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
Ref | Expression |
---|---|
suceloni | ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onelss 5683 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
2 | velsn 4141 | . . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | eqimss 3620 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴) | |
4 | 2, 3 | sylbi 206 | . . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ⊆ 𝐴) |
5 | 4 | a1i 11 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥 ⊆ 𝐴)) |
6 | 1, 5 | orim12d 879 | . . . . . . 7 ⊢ (𝐴 ∈ On → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) → (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴))) |
7 | df-suc 5646 | . . . . . . . . 9 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
8 | 7 | eleq2i 2680 | . . . . . . . 8 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴})) |
9 | elun 3715 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴})) | |
10 | 8, 9 | bitr2i 264 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴) |
11 | oridm 535 | . . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴) ↔ 𝑥 ⊆ 𝐴) | |
12 | 6, 10, 11 | 3imtr3g 283 | . . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ 𝐴)) |
13 | sssucid 5719 | . . . . . 6 ⊢ 𝐴 ⊆ suc 𝐴 | |
14 | sstr2 3575 | . . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ suc 𝐴 → 𝑥 ⊆ suc 𝐴)) | |
15 | 12, 13, 14 | syl6mpi 65 | . . . . 5 ⊢ (𝐴 ∈ On → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ suc 𝐴)) |
16 | 15 | ralrimiv 2948 | . . . 4 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴) |
17 | dftr3 4684 | . . . 4 ⊢ (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴) | |
18 | 16, 17 | sylibr 223 | . . 3 ⊢ (𝐴 ∈ On → Tr suc 𝐴) |
19 | onss 6882 | . . . . 5 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | |
20 | snssi 4280 | . . . . 5 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On) | |
21 | 19, 20 | unssd 3751 | . . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On) |
22 | 7, 21 | syl5eqss 3612 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ⊆ On) |
23 | ordon 6874 | . . . 4 ⊢ Ord On | |
24 | trssord 5657 | . . . . 5 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴) | |
25 | 24 | 3exp 1256 | . . . 4 ⊢ (Tr suc 𝐴 → (suc 𝐴 ⊆ On → (Ord On → Ord suc 𝐴))) |
26 | 23, 25 | mpii 45 | . . 3 ⊢ (Tr suc 𝐴 → (suc 𝐴 ⊆ On → Ord suc 𝐴)) |
27 | 18, 22, 26 | sylc 63 | . 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
28 | sucexg 6902 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V) | |
29 | elong 5648 | . . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴)) | |
30 | 28, 29 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (suc 𝐴 ∈ On ↔ Ord suc 𝐴)) |
31 | 27, 30 | mpbird 246 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 {csn 4125 Tr wtr 4680 Ord word 5639 Oncon0 5640 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-suc 5646 |
This theorem is referenced by: ordsuc 6906 unon 6923 onsuci 6930 ordunisuc2 6936 ordzsl 6937 onzsl 6938 tfindsg 6952 dfom2 6959 findsg 6985 tfrlem12 7372 oasuc 7491 omsuc 7493 onasuc 7495 oacl 7502 oneo 7548 omeulem1 7549 omeulem2 7550 oeordi 7554 oeworde 7560 oelim2 7562 oelimcl 7567 oeeulem 7568 oeeui 7569 oaabs2 7612 omxpenlem 7946 card2inf 8343 cantnflt 8452 cantnflem1d 8468 cnfcom 8480 r1ordg 8524 bndrank 8587 r1pw 8591 r1pwALT 8592 tcrank 8630 onssnum 8746 dfac12lem2 8849 cfsuc 8962 cfsmolem 8975 fin1a2lem1 9105 fin1a2lem2 9106 ttukeylem7 9220 alephreg 9283 gch2 9376 winainflem 9394 winalim2 9397 r1wunlim 9438 nqereu 9630 ontgval 31600 ontgsucval 31601 onsuctop 31602 sucneqond 32389 onsetreclem2 42248 |
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