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Theorem sucinc 6470
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 4433 . . 3 𝑥 ⊆ suc 𝑥
2 vex 2755 . . . 4 𝑥 ∈ V
32sucex 4516 . . . 4 suc 𝑥 ∈ V
4 suceq 4420 . . . . 5 (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
5 sucinc.1 . . . . 5 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
64, 5fvmptg 5613 . . . 4 ((𝑥 ∈ V ∧ suc 𝑥 ∈ V) → (𝐹𝑥) = suc 𝑥)
72, 3, 6mp2an 426 . . 3 (𝐹𝑥) = suc 𝑥
81, 7sseqtrri 3205 . 2 𝑥 ⊆ (𝐹𝑥)
98ax-gen 1460 1 𝑥 𝑥 ⊆ (𝐹𝑥)
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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