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Theorem sucinc2 6408
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 4350 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 ordsucss 4478 . . . . 5 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
31, 2syl 14 . . . 4 (𝐵 ∈ On → (𝐴𝐵 → suc 𝐴𝐵))
43imp 123 . . 3 ((𝐵 ∈ On ∧ 𝐴𝐵) → suc 𝐴𝐵)
5 sssucid 4390 . . 3 𝐵 ⊆ suc 𝐵
64, 5sstrdi 3152 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → suc 𝐴 ⊆ suc 𝐵)
7 onelon 4359 . . 3 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
8 elex 2735 . . . 4 (𝐴 ∈ On → 𝐴 ∈ V)
9 sucexg 4472 . . . 4 (𝐴 ∈ On → suc 𝐴 ∈ V)
10 suceq 4377 . . . . 5 (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴)
11 sucinc.1 . . . . 5 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
1210, 11fvmptg 5559 . . . 4 ((𝐴 ∈ V ∧ suc 𝐴 ∈ V) → (𝐹𝐴) = suc 𝐴)
138, 9, 12syl2anc 409 . . 3 (𝐴 ∈ On → (𝐹𝐴) = suc 𝐴)
147, 13syl 14 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐴) = suc 𝐴)
15 elex 2735 . . . 4 (𝐵 ∈ On → 𝐵 ∈ V)
16 sucexg 4472 . . . 4 (𝐵 ∈ On → suc 𝐵 ∈ V)
17 suceq 4377 . . . . 5 (𝑧 = 𝐵 → suc 𝑧 = suc 𝐵)
1817, 11fvmptg 5559 . . . 4 ((𝐵 ∈ V ∧ suc 𝐵 ∈ V) → (𝐹𝐵) = suc 𝐵)
1915, 16, 18syl2anc 409 . . 3 (𝐵 ∈ On → (𝐹𝐵) = suc 𝐵)
2019adantr 274 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐵) = suc 𝐵)
216, 14, 203sstr4d 3185 1 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1342  wcel 2135  Vcvv 2724  wss 3114  cmpt 4040  Ord word 4337  Oncon0 4338  suc csuc 4340  cfv 5185
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184  ax-un 4408
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2726  df-sbc 2950  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-opab 4041  df-mpt 4042  df-tr 4078  df-id 4268  df-iord 4341  df-on 4343  df-suc 4346  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-iota 5150  df-fun 5187  df-fv 5193
This theorem is referenced by: (None)
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