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Theorem brsuccf 33288
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 33219 . . 3 Succ = (Cup ∘ ( I ⊗ Singleton))
21breqi 5068 . 2 (𝐴Succ𝐵𝐴(Cup ∘ ( I ⊗ Singleton))𝐵)
3 brsuccf.1 . . 3 𝐴 ∈ V
4 brsuccf.2 . . 3 𝐵 ∈ V
53, 4brco 5739 . 2 (𝐴(Cup ∘ ( I ⊗ Singleton))𝐵 ↔ ∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵))
6 opex 5352 . . . . 5 𝐴, {𝐴}⟩ ∈ V
7 breq1 5065 . . . . 5 (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
86, 7ceqsexv 3546 . . . 4 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
9 snex 5327 . . . . 5 {𝐴} ∈ V
103, 9, 4brcup 33286 . . . 4 (⟨𝐴, {𝐴}⟩Cup𝐵𝐵 = (𝐴 ∪ {𝐴}))
118, 10bitri 276 . . 3 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
123brtxp2 33228 . . . . . 6 (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏))
1312anbi1i 623 . . . . 5 ((𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
14 3anass 1089 . . . . . . . . 9 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
1514anbi1i 623 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
16 an32 642 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
17 vex 3502 . . . . . . . . . . . . 13 𝑎 ∈ V
1817ideq 5721 . . . . . . . . . . . 12 (𝐴 I 𝑎𝐴 = 𝑎)
19 eqcom 2832 . . . . . . . . . . . 12 (𝐴 = 𝑎𝑎 = 𝐴)
2018, 19bitri 276 . . . . . . . . . . 11 (𝐴 I 𝑎𝑎 = 𝐴)
21 vex 3502 . . . . . . . . . . . 12 𝑏 ∈ V
223, 21brsingle 33264 . . . . . . . . . . 11 (𝐴Singleton𝑏𝑏 = {𝐴})
2320, 22anbi12i 626 . . . . . . . . . 10 ((𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑎 = 𝐴𝑏 = {𝐴}))
2423anbi1i 623 . . . . . . . . 9 (((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
25 ancom 461 . . . . . . . . 9 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ↔ ((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
26 df-3an 1083 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
2724, 25, 263bitr4i 304 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ↔ (𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
2815, 16, 273bitri 298 . . . . . . 7 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
29282exbii 1842 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
30 19.41vv 1944 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
31 opeq1 4801 . . . . . . . . 9 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
3231eqeq2d 2836 . . . . . . . 8 (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
3332anbi1d 629 . . . . . . 7 (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
34 opeq2 4802 . . . . . . . . 9 (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
3534eqeq2d 2836 . . . . . . . 8 (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
3635anbi1d 629 . . . . . . 7 (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
373, 9, 33, 36ceqsex2v 3549 . . . . . 6 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
3829, 30, 373bitr3i 302 . . . . 5 ((∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
3913, 38bitri 276 . . . 4 ((𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
4039exbii 1841 . . 3 (∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
41 df-suc 6194 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
4241eqeq2i 2838 . . 3 (𝐵 = suc 𝐴𝐵 = (𝐴 ∪ {𝐴}))
4311, 40, 423bitr4i 304 . 2 (∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)
442, 5, 433bitri 298 1 (𝐴Succ𝐵𝐵 = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  Vcvv 3499  cun 3937  {csn 4563  cop 4569   class class class wbr 5062   I cid 5457  ccom 5557  suc csuc 6190  ctxp 33177  Singletoncsingle 33185  Cupccup 33193  Succcsuccf 33195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-symdif 4222  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-eprel 5463  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-1st 7683  df-2nd 7684  df-txp 33201  df-singleton 33209  df-cup 33216  df-succf 33219
This theorem is referenced by:  dfrdg4  33298
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