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Theorem sucinc2 6057
Description: Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
Hypothesis
Ref Expression
sucinc.1 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
Assertion
Ref Expression
sucinc2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵
Allowed substitution hint:   𝐹(𝑧)

Proof of Theorem sucinc2
StepHypRef Expression
1 eloni 4140 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 ordsucss 4258 . . . . 5 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
31, 2syl 14 . . . 4 (𝐵 ∈ On → (𝐴𝐵 → suc 𝐴𝐵))
43imp 119 . . 3 ((𝐵 ∈ On ∧ 𝐴𝐵) → suc 𝐴𝐵)
5 sssucid 4180 . . 3 𝐵 ⊆ suc 𝐵
64, 5syl6ss 2985 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → suc 𝐴 ⊆ suc 𝐵)
7 onelon 4149 . . 3 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
8 elex 2583 . . . 4 (𝐴 ∈ On → 𝐴 ∈ V)
9 sucexg 4252 . . . 4 (𝐴 ∈ On → suc 𝐴 ∈ V)
10 suceq 4167 . . . . 5 (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴)
11 sucinc.1 . . . . 5 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
1210, 11fvmptg 5276 . . . 4 ((𝐴 ∈ V ∧ suc 𝐴 ∈ V) → (𝐹𝐴) = suc 𝐴)
138, 9, 12syl2anc 397 . . 3 (𝐴 ∈ On → (𝐹𝐴) = suc 𝐴)
147, 13syl 14 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐴) = suc 𝐴)
15 elex 2583 . . . 4 (𝐵 ∈ On → 𝐵 ∈ V)
16 sucexg 4252 . . . 4 (𝐵 ∈ On → suc 𝐵 ∈ V)
17 suceq 4167 . . . . 5 (𝑧 = 𝐵 → suc 𝑧 = suc 𝐵)
1817, 11fvmptg 5276 . . . 4 ((𝐵 ∈ V ∧ suc 𝐵 ∈ V) → (𝐹𝐵) = suc 𝐵)
1915, 16, 18syl2anc 397 . . 3 (𝐵 ∈ On → (𝐹𝐵) = suc 𝐵)
2019adantr 265 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐵) = suc 𝐵)
216, 14, 203sstr4d 3016 1 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  wss 2945  cmpt 3846  Ord word 4127  Oncon0 4128  suc csuc 4130  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938
This theorem is referenced by: (None)
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