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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 4332 | . . 3 ⊢ 𝑥 ⊆ suc 𝑥 | |
2 | vex 2684 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | sucex 4410 | . . . 4 ⊢ suc 𝑥 ∈ V |
4 | suceq 4319 | . . . . 5 ⊢ (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥) | |
5 | sucinc.1 | . . . . 5 ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧) | |
6 | 4, 5 | fvmptg 5490 | . . . 4 ⊢ ((𝑥 ∈ V ∧ suc 𝑥 ∈ V) → (𝐹‘𝑥) = suc 𝑥) |
7 | 2, 3, 6 | mp2an 422 | . . 3 ⊢ (𝐹‘𝑥) = suc 𝑥 |
8 | 1, 7 | sseqtrri 3127 | . 2 ⊢ 𝑥 ⊆ (𝐹‘𝑥) |
9 | 8 | ax-gen 1425 | 1 ⊢ ∀𝑥 𝑥 ⊆ (𝐹‘𝑥) |
Colors of variables: wff set class |
Syntax hints: ∀wal 1329 = wceq 1331 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 ↦ cmpt 3984 suc csuc 4282 ‘cfv 5118 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 |
This theorem is referenced by: (None) |
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