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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 7608 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | onenon 9454 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
3 | harval2 9502 | . . 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 8797 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝑥 ↔ suc 𝐴 ≼ 𝑥)) | |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ On) → (𝐴 ≺ 𝑥 ↔ suc 𝐴 ≼ 𝑥)) |
7 | peano2 7624 | . . . . . 6 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) | |
8 | nndomog 8761 | . . . . . 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 3380 | . . 3 ⊢ (𝐴 ∈ ω → {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥}) |
12 | 11 | inteqd 4842 | . 2 ⊢ (𝐴 ∈ ω → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥}) |
13 | nnon 7608 | . . 3 ⊢ (suc 𝐴 ∈ ω → suc 𝐴 ∈ On) | |
14 | intmin 4857 | . . 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 2778 | 1 ⊢ (𝐴 ∈ ω → (har‘𝐴) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {crab 3058 ⊆ wss 3844 ∩ cint 4837 class class class wbr 5031 dom cdm 5526 Oncon0 6173 suc csuc 6175 ‘cfv 6340 ωcom 7602 ≼ cdom 8556 ≺ csdm 8557 harchar 9096 cardccrd 9440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-om 7603 df-wrecs 7979 df-recs 8040 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-oi 9050 df-har 9097 df-card 9444 |
This theorem is referenced by: (None) |
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