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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 6765 | . . . . . . . 8 ⊢ (𝐵 ∈ ω → ¬ suc 𝐵 ≼ 𝐵) | |
2 | 1 | ad2antlr 481 | . . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → ¬ suc 𝐵 ≼ 𝐵) |
3 | domtr 6687 | . . . . . . . . 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 275 | . . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → (suc 𝐵 ≼ 𝐴 → suc 𝐵 ≼ 𝐵)) |
6 | 2, 5 | mtod 653 | . . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → ¬ suc 𝐵 ≼ 𝐴) |
7 | ssdomg 6680 | . . . . . . 7 ⊢ (𝐴 ∈ ω → (suc 𝐵 ⊆ 𝐴 → suc 𝐵 ≼ 𝐴)) | |
8 | 7 | ad2antrr 480 | . . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → (suc 𝐵 ⊆ 𝐴 → suc 𝐵 ≼ 𝐴)) |
9 | 6, 8 | mtod 653 | . . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → ¬ suc 𝐵 ⊆ 𝐴) |
10 | nnord 4533 | . . . . . . 7 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
11 | ordsucss 4428 | . . . . . . 7 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) | |
12 | 10, 11 | syl 14 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) |
13 | 12 | ad2antrr 480 | . . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) |
14 | 9, 13 | mtod 653 | . . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ∈ 𝐴) |
15 | nntri1 6400 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | |
16 | 15 | adantr 274 | . . . 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 6680 | . . 3 ⊢ (𝐵 ∈ ω → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | |
20 | 19 | adantl 275 | . 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 1481 ⊆ wss 3076 class class class wbr 3937 Ord word 4292 suc csuc 4295 ωcom 4512 ≼ cdom 6641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-er 6437 df-en 6643 df-dom 6644 |
This theorem is referenced by: fisbth 6785 fientri3 6811 hashennnuni 10557 fihashdom 10581 pwf1oexmid 13367 |
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