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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 4078 | . . . . 5 ⊢ (B ∈ On → Ord B) | |
2 | ordsucss 4196 | . . . . 5 ⊢ (Ord B → (A ∈ B → suc A ⊆ B)) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (B ∈ On → (A ∈ B → suc A ⊆ B)) |
4 | 3 | imp 115 | . . 3 ⊢ ((B ∈ On ∧ A ∈ B) → suc A ⊆ B) |
5 | sssucid 4118 | . . 3 ⊢ B ⊆ suc B | |
6 | 4, 5 | syl6ss 2951 | . 2 ⊢ ((B ∈ On ∧ A ∈ B) → suc A ⊆ suc B) |
7 | onelon 4087 | . . 3 ⊢ ((B ∈ On ∧ A ∈ B) → A ∈ On) | |
8 | elex 2560 | . . . 4 ⊢ (A ∈ On → A ∈ V) | |
9 | sucexg 4190 | . . . 4 ⊢ (A ∈ On → suc A ∈ V) | |
10 | suceq 4105 | . . . . 5 ⊢ (z = A → suc z = suc A) | |
11 | sucinc.1 | . . . . 5 ⊢ 𝐹 = (z ∈ V ↦ suc z) | |
12 | 10, 11 | fvmptg 5191 | . . . 4 ⊢ ((A ∈ V ∧ suc A ∈ V) → (𝐹‘A) = suc A) |
13 | 8, 9, 12 | syl2anc 391 | . . 3 ⊢ (A ∈ On → (𝐹‘A) = suc A) |
14 | 7, 13 | syl 14 | . 2 ⊢ ((B ∈ On ∧ A ∈ B) → (𝐹‘A) = suc A) |
15 | elex 2560 | . . . 4 ⊢ (B ∈ On → B ∈ V) | |
16 | sucexg 4190 | . . . 4 ⊢ (B ∈ On → suc B ∈ V) | |
17 | suceq 4105 | . . . . 5 ⊢ (z = B → suc z = suc B) | |
18 | 17, 11 | fvmptg 5191 | . . . 4 ⊢ ((B ∈ V ∧ suc B ∈ V) → (𝐹‘B) = suc B) |
19 | 15, 16, 18 | syl2anc 391 | . . 3 ⊢ (B ∈ On → (𝐹‘B) = suc B) |
20 | 19 | adantr 261 | . 2 ⊢ ((B ∈ On ∧ A ∈ B) → (𝐹‘B) = suc B) |
21 | 6, 14, 20 | 3sstr4d 2982 | 1 ⊢ ((B ∈ On ∧ A ∈ B) → (𝐹‘A) ⊆ (𝐹‘B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 ↦ cmpt 3809 Ord word 4065 Oncon0 4066 suc csuc 4068 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 |
This theorem is referenced by: (None) |
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