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Mirrors > Home > ILE Home > Th. List > nndomo | GIF version |
Description: Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.) |
Ref | Expression |
---|---|
nndomo | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | php5dom 6633 | . . . . . . . 8 ⊢ (𝐵 ∈ ω → ¬ suc 𝐵 ≼ 𝐵) | |
2 | 1 | ad2antlr 474 | . . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → ¬ suc 𝐵 ≼ 𝐵) |
3 | domtr 6556 | . . . . . . . . 9 ⊢ ((suc 𝐵 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → suc 𝐵 ≼ 𝐵) | |
4 | 3 | expcom 115 | . . . . . . . 8 ⊢ (𝐴 ≼ 𝐵 → (suc 𝐵 ≼ 𝐴 → suc 𝐵 ≼ 𝐵)) |
5 | 4 | adantl 272 | . . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → (suc 𝐵 ≼ 𝐴 → suc 𝐵 ≼ 𝐵)) |
6 | 2, 5 | mtod 625 | . . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → ¬ suc 𝐵 ≼ 𝐴) |
7 | ssdomg 6549 | . . . . . . 7 ⊢ (𝐴 ∈ ω → (suc 𝐵 ⊆ 𝐴 → suc 𝐵 ≼ 𝐴)) | |
8 | 7 | ad2antrr 473 | . . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → (suc 𝐵 ⊆ 𝐴 → suc 𝐵 ≼ 𝐴)) |
9 | 6, 8 | mtod 625 | . . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → ¬ suc 𝐵 ⊆ 𝐴) |
10 | nnord 4439 | . . . . . . 7 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
11 | ordsucss 4334 | . . . . . . 7 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) | |
12 | 10, 11 | syl 14 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) |
13 | 12 | ad2antrr 473 | . . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) |
14 | 9, 13 | mtod 625 | . . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ∈ 𝐴) |
15 | nntri1 6271 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | |
16 | 15 | adantr 271 | . . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) |
17 | 14, 16 | mpbird 166 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → 𝐴 ⊆ 𝐵) |
18 | 17 | ex 114 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≼ 𝐵 → 𝐴 ⊆ 𝐵)) |
19 | ssdomg 6549 | . . 3 ⊢ (𝐵 ∈ ω → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | |
20 | 19 | adantl 272 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
21 | 18, 20 | impbid 128 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1439 ⊆ wss 3000 class class class wbr 3851 Ord word 4198 suc csuc 4201 ωcom 4418 ≼ cdom 6510 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-er 6306 df-en 6512 df-dom 6513 |
This theorem is referenced by: fisbth 6653 fientri3 6679 hashennnuni 10248 fihashdom 10272 |
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