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Theorem refun0 22749
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 2737 . . . 4 𝐴 = 𝐴
2 eqid 2737 . . . 4 𝐵 = 𝐵
31, 2refbas 22744 . . 3 (𝐴Ref𝐵 𝐵 = 𝐴)
43adantr 481 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → 𝐵 = 𝐴)
5 elun 4094 . . . 4 (𝑥 ∈ (𝐴 ∪ {∅}) ↔ (𝑥𝐴𝑥 ∈ {∅}))
6 refssex 22745 . . . . . 6 ((𝐴Ref𝐵𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
76adantlr 712 . . . . 5 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
8 0ss 4341 . . . . . . . . 9 ∅ ⊆ 𝑦
98a1i 11 . . . . . . . 8 ((𝐴Ref𝐵𝑦𝐵) → ∅ ⊆ 𝑦)
109reximdva0 4296 . . . . . . 7 ((𝐴Ref𝐵𝐵 ≠ ∅) → ∃𝑦𝐵 ∅ ⊆ 𝑦)
1110adantr 481 . . . . . 6 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦𝐵 ∅ ⊆ 𝑦)
12 elsni 4588 . . . . . . . 8 (𝑥 ∈ {∅} → 𝑥 = ∅)
13 sseq1 3956 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ⊆ 𝑦))
1413rexbidv 3172 . . . . . . . 8 (𝑥 = ∅ → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 ∅ ⊆ 𝑦))
1512, 14syl 17 . . . . . . 7 (𝑥 ∈ {∅} → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 ∅ ⊆ 𝑦))
1615adantl 482 . . . . . 6 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 ∅ ⊆ 𝑦))
1711, 16mpbird 256 . . . . 5 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦𝐵 𝑥𝑦)
187, 17jaodan 955 . . . 4 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥 ∈ {∅})) → ∃𝑦𝐵 𝑥𝑦)
195, 18sylan2b 594 . . 3 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ (𝐴 ∪ {∅})) → ∃𝑦𝐵 𝑥𝑦)
2019ralrimiva 3140 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦𝐵 𝑥𝑦)
21 refrel 22742 . . . . . 6 Rel Ref
2221brrelex1i 5662 . . . . 5 (𝐴Ref𝐵𝐴 ∈ V)
23 p0ex 5322 . . . . 5 {∅} ∈ V
24 unexg 7641 . . . . 5 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ∪ {∅}) ∈ V)
2522, 23, 24sylancl 586 . . . 4 (𝐴Ref𝐵 → (𝐴 ∪ {∅}) ∈ V)
26 uniun 4876 . . . . . 6 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
27 0ex 5246 . . . . . . . 8 ∅ ∈ V
2827unisn 4872 . . . . . . 7 {∅} = ∅
2928uneq2i 4105 . . . . . 6 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
30 un0 4335 . . . . . 6 ( 𝐴 ∪ ∅) = 𝐴
3126, 29, 303eqtrri 2770 . . . . 5 𝐴 = (𝐴 ∪ {∅})
3231, 2isref 22743 . . . 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 481 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → ((𝐴 ∪ {∅})Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦𝐵 𝑥𝑦)))
354, 20, 34mpbir2and 710 1 ((𝐴Ref𝐵𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1540  wcel 2105  wne 2941  wral 3062  wrex 3071  Vcvv 3441  cun 3895  wss 3897  c0 4267  {csn 4571   cuni 4850   class class class wbr 5087  Refcref 22736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-xp 5614  df-rel 5615  df-ref 22739
This theorem is referenced by:  locfinref  31931
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