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Mirrors > Home > MPE Home > Th. List > nndomog | Structured version Visualization version GIF version |
Description: Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 9173 when both are natural numbers. (Contributed by NM, 17-Jun-1998.) Generalize from nndomo 9173. (Revised by RP, 5-Nov-2023.) Avoid ax-pow 5318. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
nndomog | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnfi 9107 | . . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
2 | domnsymfi 9143 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴) | |
3 | 1, 2 | sylan 580 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
4 | 3 | ex 413 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴)) |
5 | php2 9151 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) | |
6 | 5 | ex 413 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴)) |
7 | 4, 6 | nsyld 156 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ≼ 𝐵 → ¬ 𝐵 ⊊ 𝐴)) |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 → ¬ 𝐵 ⊊ 𝐴)) |
9 | nnord 7806 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
10 | eloni 6325 | . . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
11 | ordtri1 6348 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | |
12 | ordelpss 6343 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴)) | |
13 | 12 | ancoms 459 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴)) |
14 | 13 | notbid 317 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ⊊ 𝐴)) |
15 | 11, 14 | bitrd 278 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ⊊ 𝐴)) |
16 | 9, 10, 15 | syl2an 596 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ⊊ 𝐴)) |
17 | 8, 16 | sylibrd 258 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 → 𝐴 ⊆ 𝐵)) |
18 | ssdomfi2 9140 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → 𝐴 ≼ 𝐵) | |
19 | 18 | 3expia 1121 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
20 | 1, 19 | sylan 580 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
21 | 17, 20 | impbid 211 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ⊆ wss 3908 ⊊ wpss 3909 class class class wbr 5103 Ord word 6314 Oncon0 6315 ωcom 7798 ≼ cdom 8877 ≺ csdm 8878 Fincfn 8879 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-om 7799 df-1o 8408 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 |
This theorem is referenced by: onomeneq 9168 nndomo 9173 harsucnn 9930 |
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