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Theorem brsuccf 35923
Description: Binary relation form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brsuccf.1 𝐴 ∈ V
brsuccf.2 𝐵 ∈ V
Assertion
Ref Expression
brsuccf (𝐴Succ𝐵𝐵 = suc 𝐴)

Proof of Theorem brsuccf
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-succf 35854 . . 3 Succ = (Cup ∘ ( I ⊗ Singleton))
21breqi 5154 . 2 (𝐴Succ𝐵𝐴(Cup ∘ ( I ⊗ Singleton))𝐵)
3 brsuccf.1 . . 3 𝐴 ∈ V
4 brsuccf.2 . . 3 𝐵 ∈ V
53, 4brco 5884 . 2 (𝐴(Cup ∘ ( I ⊗ Singleton))𝐵 ↔ ∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵))
6 opex 5475 . . . . 5 𝐴, {𝐴}⟩ ∈ V
7 breq1 5151 . . . . 5 (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
86, 7ceqsexv 3530 . . . 4 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
9 snex 5442 . . . . 5 {𝐴} ∈ V
103, 9, 4brcup 35921 . . . 4 (⟨𝐴, {𝐴}⟩Cup𝐵𝐵 = (𝐴 ∪ {𝐴}))
118, 10bitri 275 . . 3 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
123brtxp2 35863 . . . . . 6 (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏))
1312anbi1i 624 . . . . 5 ((𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
14 3anass 1094 . . . . . . . . 9 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
1514anbi1i 624 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
16 an32 646 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
17 vex 3482 . . . . . . . . . . . . 13 𝑎 ∈ V
1817ideq 5866 . . . . . . . . . . . 12 (𝐴 I 𝑎𝐴 = 𝑎)
19 eqcom 2742 . . . . . . . . . . . 12 (𝐴 = 𝑎𝑎 = 𝐴)
2018, 19bitri 275 . . . . . . . . . . 11 (𝐴 I 𝑎𝑎 = 𝐴)
21 vex 3482 . . . . . . . . . . . 12 𝑏 ∈ V
223, 21brsingle 35899 . . . . . . . . . . 11 (𝐴Singleton𝑏𝑏 = {𝐴})
2320, 22anbi12i 628 . . . . . . . . . 10 ((𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑎 = 𝐴𝑏 = {𝐴}))
2423anbi1i 624 . . . . . . . . 9 (((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
25 ancom 460 . . . . . . . . 9 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ↔ ((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
26 df-3an 1088 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
2724, 25, 263bitr4i 303 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ↔ (𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
2815, 16, 273bitri 297 . . . . . . 7 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
29282exbii 1846 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
30 19.41vv 1948 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
31 opeq1 4878 . . . . . . . . 9 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
3231eqeq2d 2746 . . . . . . . 8 (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
3332anbi1d 631 . . . . . . 7 (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
34 opeq2 4879 . . . . . . . . 9 (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
3534eqeq2d 2746 . . . . . . . 8 (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
3635anbi1d 631 . . . . . . 7 (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
373, 9, 33, 36ceqsex2v 3536 . . . . . 6 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
3829, 30, 373bitr3i 301 . . . . 5 ((∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
3913, 38bitri 275 . . . 4 ((𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
4039exbii 1845 . . 3 (∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
41 df-suc 6392 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
4241eqeq2i 2748 . . 3 (𝐵 = suc 𝐴𝐵 = (𝐴 ∪ {𝐴}))
4311, 40, 423bitr4i 303 . 2 (∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)
442, 5, 433bitri 297 1 (𝐴Succ𝐵𝐵 = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cun 3961  {csn 4631  cop 4637   class class class wbr 5148   I cid 5582  ccom 5693  suc csuc 6388  ctxp 35812  Singletoncsingle 35820  Cupccup 35828  Succcsuccf 35830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-singleton 35844  df-cup 35851  df-succf 35854
This theorem is referenced by:  dfrdg4  35933
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