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Theorem invdisjrab 5130
Description: The restricted class abstractions {𝑥𝐵𝐶 = 𝑦} for distinct 𝑦𝐴 are disjoint. (Contributed by AV, 6-May-2020.) (Proof shortened by GG, 26-Jan-2024.)
Assertion
Ref Expression
invdisjrab Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
Distinct variable groups:   𝑥,𝐵   𝑦,𝐶   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥)

Proof of Theorem invdisjrab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2905 . . . . 5 𝑥𝑧
2 nfcv 2905 . . . . 5 𝑥𝐵
3 nfcsb1v 3923 . . . . . 6 𝑥𝑧 / 𝑥𝐶
43nfeq1 2921 . . . . 5 𝑥𝑧 / 𝑥𝐶 = 𝑦
5 csbeq1a 3913 . . . . . 6 (𝑥 = 𝑧𝐶 = 𝑧 / 𝑥𝐶)
65eqeq1d 2739 . . . . 5 (𝑥 = 𝑧 → (𝐶 = 𝑦𝑧 / 𝑥𝐶 = 𝑦))
71, 2, 4, 6elrabf 3688 . . . 4 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} ↔ (𝑧𝐵𝑧 / 𝑥𝐶 = 𝑦))
8 simprr 773 . . . 4 ((𝑦𝐴 ∧ (𝑧𝐵𝑧 / 𝑥𝐶 = 𝑦)) → 𝑧 / 𝑥𝐶 = 𝑦)
97, 8sylan2b 594 . . 3 ((𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}) → 𝑧 / 𝑥𝐶 = 𝑦)
109rgen2 3199 . 2 𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦
11 invdisj 5129 . 2 (∀𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦})
1210, 11ax-mp 5 1 Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  csb 3899  Disj wdisj 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rmo 3380  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-disj 5111
This theorem is referenced by:  disjwrdpfx  14738
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