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Mirrors > Home > ILE Home > Th. List > sucinc2 | GIF version |
Description: Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.) |
Ref | Expression |
---|---|
sucinc.1 | ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧) |
Ref | Expression |
---|---|
sucinc2 | ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4350 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | ordsucss 4478 | . . . . 5 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
4 | 3 | imp 123 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ 𝐵) |
5 | sssucid 4390 | . . 3 ⊢ 𝐵 ⊆ suc 𝐵 | |
6 | 4, 5 | sstrdi 3152 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ suc 𝐵) |
7 | onelon 4359 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On) | |
8 | elex 2735 | . . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
9 | sucexg 4472 | . . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V) | |
10 | suceq 4377 | . . . . 5 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴) | |
11 | sucinc.1 | . . . . 5 ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧) | |
12 | 10, 11 | fvmptg 5559 | . . . 4 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 ∈ V) → (𝐹‘𝐴) = suc 𝐴) |
13 | 8, 9, 12 | syl2anc 409 | . . 3 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = suc 𝐴) |
14 | 7, 13 | syl 14 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = suc 𝐴) |
15 | elex 2735 | . . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ V) | |
16 | sucexg 4472 | . . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ V) | |
17 | suceq 4377 | . . . . 5 ⊢ (𝑧 = 𝐵 → suc 𝑧 = suc 𝐵) | |
18 | 17, 11 | fvmptg 5559 | . . . 4 ⊢ ((𝐵 ∈ V ∧ suc 𝐵 ∈ V) → (𝐹‘𝐵) = suc 𝐵) |
19 | 15, 16, 18 | syl2anc 409 | . . 3 ⊢ (𝐵 ∈ On → (𝐹‘𝐵) = suc 𝐵) |
20 | 19 | adantr 274 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐵) = suc 𝐵) |
21 | 6, 14, 20 | 3sstr4d 3185 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2724 ⊆ wss 3114 ↦ cmpt 4040 Ord word 4337 Oncon0 4338 suc csuc 4340 ‘cfv 5185 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2726 df-sbc 2950 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-iord 4341 df-on 4343 df-suc 4346 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-iota 5150 df-fun 5187 df-fv 5193 |
This theorem is referenced by: (None) |
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