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Mirrors > Home > MPE Home > Th. List > harsucnn | Structured version Visualization version GIF version |
Description: The next cardinal after a finite ordinal is the successor ordinal. (Contributed by RP, 5-Nov-2023.) |
Ref | Expression |
---|---|
harsucnn | ⊢ (𝐴 ∈ ω → (har‘𝐴) = suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7566 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | onenon 9362 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
3 | harval2 9410 | . . 3 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ω → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) |
5 | sucdom 8699 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝑥 ↔ suc 𝐴 ≼ 𝑥)) | |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ On) → (𝐴 ≺ 𝑥 ↔ suc 𝐴 ≼ 𝑥)) |
7 | peano2 7582 | . . . . . 6 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) | |
8 | nndomog 8692 | . . . . . 6 ⊢ ((suc 𝐴 ∈ ω ∧ 𝑥 ∈ On) → (suc 𝐴 ≼ 𝑥 ↔ suc 𝐴 ⊆ 𝑥)) | |
9 | 7, 8 | sylan 583 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ On) → (suc 𝐴 ≼ 𝑥 ↔ suc 𝐴 ⊆ 𝑥)) |
10 | 6, 9 | bitrd 282 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ On) → (𝐴 ≺ 𝑥 ↔ suc 𝐴 ⊆ 𝑥)) |
11 | 10 | rabbidva 3425 | . . 3 ⊢ (𝐴 ∈ ω → {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥}) |
12 | 11 | inteqd 4843 | . 2 ⊢ (𝐴 ∈ ω → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥}) |
13 | nnon 7566 | . . 3 ⊢ (suc 𝐴 ∈ ω → suc 𝐴 ∈ On) | |
14 | intmin 4858 | . . 3 ⊢ (suc 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴) | |
15 | 7, 13, 14 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ω → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴) |
16 | 4, 12, 15 | 3eqtrd 2837 | 1 ⊢ (𝐴 ∈ ω → (har‘𝐴) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 ∩ cint 4838 class class class wbr 5030 dom cdm 5519 Oncon0 6159 suc csuc 6161 ‘cfv 6324 ωcom 7560 ≼ cdom 8490 ≺ csdm 8491 harchar 9004 cardccrd 9348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-om 7561 df-wrecs 7930 df-recs 7991 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-oi 8958 df-har 9005 df-card 9352 |
This theorem is referenced by: (None) |
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