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Theorem refun0 22123
Description: Adding the empty set preserves refinements. (Contributed by Thierry Arnoux, 31-Jan-2020.)
Assertion
Ref Expression
refun0 ((𝐴Ref𝐵𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)

Proof of Theorem refun0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . . 4 𝐴 = 𝐴
2 eqid 2824 . . . 4 𝐵 = 𝐵
31, 2refbas 22118 . . 3 (𝐴Ref𝐵 𝐵 = 𝐴)
43adantr 484 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → 𝐵 = 𝐴)
5 elun 4111 . . . 4 (𝑥 ∈ (𝐴 ∪ {∅}) ↔ (𝑥𝐴𝑥 ∈ {∅}))
6 refssex 22119 . . . . . 6 ((𝐴Ref𝐵𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
76adantlr 714 . . . . 5 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
8 0ss 4333 . . . . . . . . 9 ∅ ⊆ 𝑦
98a1i 11 . . . . . . . 8 ((𝐴Ref𝐵𝑦𝐵) → ∅ ⊆ 𝑦)
109reximdva0 4295 . . . . . . 7 ((𝐴Ref𝐵𝐵 ≠ ∅) → ∃𝑦𝐵 ∅ ⊆ 𝑦)
1110adantr 484 . . . . . 6 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦𝐵 ∅ ⊆ 𝑦)
12 elsni 4567 . . . . . . . 8 (𝑥 ∈ {∅} → 𝑥 = ∅)
13 sseq1 3978 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ⊆ 𝑦))
1413rexbidv 3289 . . . . . . . 8 (𝑥 = ∅ → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 ∅ ⊆ 𝑦))
1512, 14syl 17 . . . . . . 7 (𝑥 ∈ {∅} → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 ∅ ⊆ 𝑦))
1615adantl 485 . . . . . 6 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 ∅ ⊆ 𝑦))
1711, 16mpbird 260 . . . . 5 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦𝐵 𝑥𝑦)
187, 17jaodan 955 . . . 4 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥 ∈ {∅})) → ∃𝑦𝐵 𝑥𝑦)
195, 18sylan2b 596 . . 3 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ (𝐴 ∪ {∅})) → ∃𝑦𝐵 𝑥𝑦)
2019ralrimiva 3177 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦𝐵 𝑥𝑦)
21 refrel 22116 . . . . . 6 Rel Ref
2221brrelex1i 5595 . . . . 5 (𝐴Ref𝐵𝐴 ∈ V)
23 p0ex 5272 . . . . 5 {∅} ∈ V
24 unexg 7466 . . . . 5 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ∪ {∅}) ∈ V)
2522, 23, 24sylancl 589 . . . 4 (𝐴Ref𝐵 → (𝐴 ∪ {∅}) ∈ V)
26 uniun 4847 . . . . . 6 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
27 0ex 5197 . . . . . . . 8 ∅ ∈ V
2827unisn 4844 . . . . . . 7 {∅} = ∅
2928uneq2i 4122 . . . . . 6 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
30 un0 4327 . . . . . 6 ( 𝐴 ∪ ∅) = 𝐴
3126, 29, 303eqtrri 2852 . . . . 5 𝐴 = (𝐴 ∪ {∅})
3231, 2isref 22117 . . . 4 ((𝐴 ∪ {∅}) ∈ V → ((𝐴 ∪ {∅})Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦𝐵 𝑥𝑦)))
3325, 32syl 17 . . 3 (𝐴Ref𝐵 → ((𝐴 ∪ {∅})Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦𝐵 𝑥𝑦)))
3433adantr 484 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → ((𝐴 ∪ {∅})Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦𝐵 𝑥𝑦)))
354, 20, 34mpbir2and 712 1 ((𝐴Ref𝐵𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  Vcvv 3480  cun 3917  wss 3919  c0 4276  {csn 4550   cuni 4824   class class class wbr 5052  Refcref 22110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-ref 22113
This theorem is referenced by:  locfinref  31165
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