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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 4433 | . . 3 ⊢ 𝑥 ⊆ suc 𝑥 | |
2 | vex 2755 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | sucex 4516 | . . . 4 ⊢ suc 𝑥 ∈ V |
4 | suceq 4420 | . . . . 5 ⊢ (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥) | |
5 | sucinc.1 | . . . . 5 ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧) | |
6 | 4, 5 | fvmptg 5613 | . . . 4 ⊢ ((𝑥 ∈ V ∧ suc 𝑥 ∈ V) → (𝐹‘𝑥) = suc 𝑥) |
7 | 2, 3, 6 | mp2an 426 | . . 3 ⊢ (𝐹‘𝑥) = suc 𝑥 |
8 | 1, 7 | sseqtrri 3205 | . 2 ⊢ 𝑥 ⊆ (𝐹‘𝑥) |
9 | 8 | ax-gen 1460 | 1 ⊢ ∀𝑥 𝑥 ⊆ (𝐹‘𝑥) |
Colors of variables: wff set class |
Syntax hints: ∀wal 1362 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 ↦ cmpt 4079 suc csuc 4383 ‘cfv 5235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-suc 4389 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 |
This theorem is referenced by: (None) |
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