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Theorem brsuccf 32173
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 32104 . . 3 Succ = (Cup ∘ ( I ⊗ Singleton))
21breqi 4691 . 2 (𝐴Succ𝐵𝐴(Cup ∘ ( I ⊗ Singleton))𝐵)
3 brsuccf.1 . . 3 𝐴 ∈ V
4 brsuccf.2 . . 3 𝐵 ∈ V
53, 4brco 5325 . 2 (𝐴(Cup ∘ ( I ⊗ Singleton))𝐵 ↔ ∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵))
6 opex 4962 . . . . 5 𝐴, {𝐴}⟩ ∈ V
7 breq1 4688 . . . . 5 (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
86, 7ceqsexv 3273 . . . 4 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
9 snex 4938 . . . . 5 {𝐴} ∈ V
103, 9, 4brcup 32171 . . . 4 (⟨𝐴, {𝐴}⟩Cup𝐵𝐵 = (𝐴 ∪ {𝐴}))
118, 10bitri 264 . . 3 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
123brtxp2 32113 . . . . . 6 (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏))
1312anbi1i 731 . . . . 5 ((𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
14 3anass 1059 . . . . . . . . 9 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
1514anbi1i 731 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
16 an32 856 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
17 vex 3234 . . . . . . . . . . . . 13 𝑎 ∈ V
1817ideq 5307 . . . . . . . . . . . 12 (𝐴 I 𝑎𝐴 = 𝑎)
19 eqcom 2658 . . . . . . . . . . . 12 (𝐴 = 𝑎𝑎 = 𝐴)
2018, 19bitri 264 . . . . . . . . . . 11 (𝐴 I 𝑎𝑎 = 𝐴)
21 vex 3234 . . . . . . . . . . . 12 𝑏 ∈ V
223, 21brsingle 32149 . . . . . . . . . . 11 (𝐴Singleton𝑏𝑏 = {𝐴})
2320, 22anbi12i 733 . . . . . . . . . 10 ((𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑎 = 𝐴𝑏 = {𝐴}))
2423anbi1i 731 . . . . . . . . 9 (((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
25 ancom 465 . . . . . . . . 9 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ↔ ((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
26 df-3an 1056 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
2724, 25, 263bitr4i 292 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ↔ (𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
2815, 16, 273bitri 286 . . . . . . 7 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
29282exbii 1815 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
30 19.41vv 1918 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
31 opeq1 4433 . . . . . . . . 9 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
3231eqeq2d 2661 . . . . . . . 8 (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
3332anbi1d 741 . . . . . . 7 (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
34 opeq2 4434 . . . . . . . . 9 (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
3534eqeq2d 2661 . . . . . . . 8 (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
3635anbi1d 741 . . . . . . 7 (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
373, 9, 33, 36ceqsex2v 3276 . . . . . 6 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
3829, 30, 373bitr3i 290 . . . . 5 ((∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
3913, 38bitri 264 . . . 4 ((𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
4039exbii 1814 . . 3 (∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
41 df-suc 5767 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
4241eqeq2i 2663 . . 3 (𝐵 = suc 𝐴𝐵 = (𝐴 ∪ {𝐴}))
4311, 40, 423bitr4i 292 . 2 (∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)
442, 5, 433bitri 286 1 (𝐴Succ𝐵𝐵 = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cun 3605  {csn 4210  cop 4216   class class class wbr 4685   I cid 5052  ccom 5147  suc csuc 5763  ctxp 32062  Singletoncsingle 32070  Cupccup 32078  Succcsuccf 32080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-symdif 3877  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-eprel 5058  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211  df-txp 32086  df-singleton 32094  df-cup 32101  df-succf 32104
This theorem is referenced by:  dfrdg4  32183
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