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Theorem sucinc 6271
Description: Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
Hypothesis
Ref Expression
sucinc.1 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
Assertion
Ref Expression
sucinc 𝑥 𝑥 ⊆ (𝐹𝑥)
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑧)

Proof of Theorem sucinc
StepHypRef Expression
1 sssucid 4275 . . 3 𝑥 ⊆ suc 𝑥
2 vex 2644 . . . 4 𝑥 ∈ V
32sucex 4353 . . . 4 suc 𝑥 ∈ V
4 suceq 4262 . . . . 5 (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
5 sucinc.1 . . . . 5 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
64, 5fvmptg 5429 . . . 4 ((𝑥 ∈ V ∧ suc 𝑥 ∈ V) → (𝐹𝑥) = suc 𝑥)
72, 3, 6mp2an 420 . . 3 (𝐹𝑥) = suc 𝑥
81, 7sseqtr4i 3082 . 2 𝑥 ⊆ (𝐹𝑥)
98ax-gen 1393 1 𝑥 𝑥 ⊆ (𝐹𝑥)
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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