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Mirrors > Home > ILE Home > Th. List > sucinc | GIF version |
Description: Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.) |
Ref | Expression |
---|---|
sucinc.1 | ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧) |
Ref | Expression |
---|---|
sucinc | ⊢ ∀𝑥 𝑥 ⊆ (𝐹‘𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssucid 4275 | . . 3 ⊢ 𝑥 ⊆ suc 𝑥 | |
2 | vex 2644 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | sucex 4353 | . . . 4 ⊢ suc 𝑥 ∈ V |
4 | suceq 4262 | . . . . 5 ⊢ (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥) | |
5 | sucinc.1 | . . . . 5 ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧) | |
6 | 4, 5 | fvmptg 5429 | . . . 4 ⊢ ((𝑥 ∈ V ∧ suc 𝑥 ∈ V) → (𝐹‘𝑥) = suc 𝑥) |
7 | 2, 3, 6 | mp2an 420 | . . 3 ⊢ (𝐹‘𝑥) = suc 𝑥 |
8 | 1, 7 | sseqtr4i 3082 | . 2 ⊢ 𝑥 ⊆ (𝐹‘𝑥) |
9 | 8 | ax-gen 1393 | 1 ⊢ ∀𝑥 𝑥 ⊆ (𝐹‘𝑥) |
Colors of variables: wff set class |
Syntax hints: ∀wal 1297 = wceq 1299 ∈ wcel 1448 Vcvv 2641 ⊆ wss 3021 ↦ cmpt 3929 suc csuc 4225 ‘cfv 5059 |
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-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-suc 4231 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 |
This theorem is referenced by: (None) |
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