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Theorem brsuccf 31682
Description: Binary relationship 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 31612 . . 3 Succ = (Cup ∘ ( I ⊗ Singleton))
21breqi 4624 . 2 (𝐴Succ𝐵𝐴(Cup ∘ ( I ⊗ Singleton))𝐵)
3 brsuccf.1 . . 3 𝐴 ∈ V
4 brsuccf.2 . . 3 𝐵 ∈ V
53, 4brco 5257 . 2 (𝐴(Cup ∘ ( I ⊗ Singleton))𝐵 ↔ ∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵))
6 opex 4898 . . . . 5 𝐴, {𝐴}⟩ ∈ V
7 breq1 4621 . . . . 5 (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
86, 7ceqsexv 3233 . . . 4 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
9 snex 4874 . . . . 5 {𝐴} ∈ V
103, 9, 4brcup 31680 . . . 4 (⟨𝐴, {𝐴}⟩Cup𝐵𝐵 = (𝐴 ∪ {𝐴}))
118, 10bitri 264 . . 3 (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
123brtxp2 31622 . . . . . 6 (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏))
1312anbi1i 730 . . . . 5 ((𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
14 3anass 1040 . . . . . . . . 9 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
1514anbi1i 730 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
16 an32 838 . . . . . . . 8 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)))
17 vex 3194 . . . . . . . . . . . . 13 𝑎 ∈ V
1817ideq 5239 . . . . . . . . . . . 12 (𝐴 I 𝑎𝐴 = 𝑎)
19 eqcom 2633 . . . . . . . . . . . 12 (𝐴 = 𝑎𝑎 = 𝐴)
2018, 19bitri 264 . . . . . . . . . . 11 (𝐴 I 𝑎𝑎 = 𝐴)
21 vex 3194 . . . . . . . . . . . 12 𝑏 ∈ V
223, 21brsingle 31658 . . . . . . . . . . 11 (𝐴Singleton𝑏𝑏 = {𝐴})
2320, 22anbi12i 732 . . . . . . . . . 10 ((𝐴 I 𝑎𝐴Singleton𝑏) ↔ (𝑎 = 𝐴𝑏 = {𝐴}))
2423anbi1i 730 . . . . . . . . 9 (((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
25 ancom 466 . . . . . . . . 9 (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎𝐴Singleton𝑏)) ↔ ((𝐴 I 𝑎𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
26 df-3an 1038 . . . . . . . . 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 1773 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
30 19.41vv 1917 . . . . . 6 (∃𝑎𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
31 opeq1 4375 . . . . . . . . 9 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
3231eqeq2d 2636 . . . . . . . 8 (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
3332anbi1d 740 . . . . . . 7 (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
34 opeq2 4376 . . . . . . . . 9 (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
3534eqeq2d 2636 . . . . . . . 8 (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
3635anbi1d 740 . . . . . . 7 (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
373, 9, 33, 36ceqsex2v 3236 . . . . . 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 1772 . . 3 (∃𝑥(𝐴( I ⊗ Singleton)𝑥𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
41 df-suc 5691 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
4241eqeq2i 2638 . . 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1992  Vcvv 3191  cun 3558  {csn 4153  cop 4159   class class class wbr 4618   I cid 4989  ccom 5083  suc csuc 5687  ctxp 31570  Singletoncsingle 31578  Cupccup 31586  Succcsuccf 31588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-symdif 3827  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fo 5856  df-fv 5858  df-1st 7116  df-2nd 7117  df-txp 31594  df-singleton 31602  df-cup 31609  df-succf 31612
This theorem is referenced by:  dfrdg4  31692
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