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Theorem brsuccf 32495
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 32426 . . 3 Succ = (Cup ∘ ( I ⊗ Singleton))
21breqi 4817 . 2 (𝐴Succ𝐵𝐴(Cup ∘ ( I ⊗ Singleton))𝐵)
3 brsuccf.1 . . 3 𝐴 ∈ V
4 brsuccf.2 . . 3 𝐵 ∈ V
53, 4brco 5463 . 2 (𝐴(Cup ∘ ( I ⊗ Singleton))𝐵 ↔ ∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵))
6 opex 5090 . . . . 5 𝐴, {𝐴}⟩ ∈ V
7 breq1 4814 . . . . 5 (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
86, 7ceqsexv 3395 . . . 4 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
9 snex 5066 . . . . 5 {𝐴} ∈ V
103, 9, 4brcup 32493 . . . 4 (⟨𝐴, {𝐴}⟩Cup𝐵𝐵 = (𝐴 ∪ {𝐴}))
118, 10bitri 266 . . 3 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
123brtxp2 32435 . . . . . 6 (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏))
1312anbi1i 617 . . . . 5 ((𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
14 3anass 1116 . . . . . . . . 9 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
1514anbi1i 617 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
16 an32 636 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
17 vex 3353 . . . . . . . . . . . . 13 𝑎 ∈ V
1817ideq 5445 . . . . . . . . . . . 12 (𝐴 I 𝑎𝐴 = 𝑎)
19 eqcom 2772 . . . . . . . . . . . 12 (𝐴 = 𝑎𝑎 = 𝐴)
2018, 19bitri 266 . . . . . . . . . . 11 (𝐴 I 𝑎𝑎 = 𝐴)
21 vex 3353 . . . . . . . . . . . 12 𝑏 ∈ V
223, 21brsingle 32471 . . . . . . . . . . 11 (𝐴Singleton𝑏𝑏 = {𝐴})
2320, 22anbi12i 620 . . . . . . . . . 10 ((𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑎 = 𝐴𝑏 = {𝐴}))
2423anbi1i 617 . . . . . . . . 9 (((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
25 ancom 452 . . . . . . . . 9 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ↔ ((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
26 df-3an 1109 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
2724, 25, 263bitr4i 294 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ↔ (𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
2815, 16, 273bitri 288 . . . . . . 7 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
29282exbii 1944 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
30 19.41vv 2045 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
31 opeq1 4561 . . . . . . . . 9 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
3231eqeq2d 2775 . . . . . . . 8 (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
3332anbi1d 623 . . . . . . 7 (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
34 opeq2 4562 . . . . . . . . 9 (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
3534eqeq2d 2775 . . . . . . . 8 (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
3635anbi1d 623 . . . . . . 7 (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
373, 9, 33, 36ceqsex2v 3398 . . . . . 6 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
3829, 30, 373bitr3i 292 . . . . 5 ((∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
3913, 38bitri 266 . . . 4 ((𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
4039exbii 1943 . . 3 (∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
41 df-suc 5916 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
4241eqeq2i 2777 . . 3 (𝐵 = suc 𝐴𝐵 = (𝐴 ∪ {𝐴}))
4311, 40, 423bitr4i 294 . 2 (∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)
442, 5, 433bitri 288 1 (𝐴Succ𝐵𝐵 = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  cun 3732  {csn 4336  cop 4342   class class class wbr 4811   I cid 5186  ccom 5283  suc csuc 5912  ctxp 32384  Singletoncsingle 32392  Cupccup 32400  Succcsuccf 32402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-symdif 4007  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-eprel 5192  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fo 6076  df-fv 6078  df-1st 7368  df-2nd 7369  df-txp 32408  df-singleton 32416  df-cup 32423  df-succf 32426
This theorem is referenced by:  dfrdg4  32505
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