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Theorem refun0 22889
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 2733 . . . 4 𝐴 = 𝐴
2 eqid 2733 . . . 4 𝐵 = 𝐵
31, 2refbas 22884 . . 3 (𝐴Ref𝐵 𝐵 = 𝐴)
43adantr 482 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → 𝐵 = 𝐴)
5 elun 4112 . . . 4 (𝑥 ∈ (𝐴 ∪ {∅}) ↔ (𝑥𝐴𝑥 ∈ {∅}))
6 refssex 22885 . . . . . 6 ((𝐴Ref𝐵𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
76adantlr 714 . . . . 5 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
8 0ss 4360 . . . . . . . . 9 ∅ ⊆ 𝑦
98a1i 11 . . . . . . . 8 ((𝐴Ref𝐵𝑦𝐵) → ∅ ⊆ 𝑦)
109reximdva0 4315 . . . . . . 7 ((𝐴Ref𝐵𝐵 ≠ ∅) → ∃𝑦𝐵 ∅ ⊆ 𝑦)
1110adantr 482 . . . . . 6 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦𝐵 ∅ ⊆ 𝑦)
12 elsni 4607 . . . . . . . 8 (𝑥 ∈ {∅} → 𝑥 = ∅)
13 sseq1 3973 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ⊆ 𝑦))
1413rexbidv 3172 . . . . . . . 8 (𝑥 = ∅ → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 ∅ ⊆ 𝑦))
1512, 14syl 17 . . . . . . 7 (𝑥 ∈ {∅} → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 ∅ ⊆ 𝑦))
1615adantl 483 . . . . . 6 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 ∅ ⊆ 𝑦))
1711, 16mpbird 257 . . . . 5 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦𝐵 𝑥𝑦)
187, 17jaodan 957 . . . 4 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥 ∈ {∅})) → ∃𝑦𝐵 𝑥𝑦)
195, 18sylan2b 595 . . 3 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ (𝐴 ∪ {∅})) → ∃𝑦𝐵 𝑥𝑦)
2019ralrimiva 3140 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦𝐵 𝑥𝑦)
21 refrel 22882 . . . . . 6 Rel Ref
2221brrelex1i 5692 . . . . 5 (𝐴Ref𝐵𝐴 ∈ V)
23 p0ex 5343 . . . . 5 {∅} ∈ V
24 unexg 7687 . . . . 5 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ∪ {∅}) ∈ V)
2522, 23, 24sylancl 587 . . . 4 (𝐴Ref𝐵 → (𝐴 ∪ {∅}) ∈ V)
26 uniun 4895 . . . . . 6 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
27 0ex 5268 . . . . . . . 8 ∅ ∈ V
2827unisn 4891 . . . . . . 7 {∅} = ∅
2928uneq2i 4124 . . . . . 6 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
30 un0 4354 . . . . . 6 ( 𝐴 ∪ ∅) = 𝐴
3126, 29, 303eqtrri 2766 . . . . 5 𝐴 = (𝐴 ∪ {∅})
3231, 2isref 22883 . . . 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 482 . 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 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  Vcvv 3447  cun 3912  wss 3914  c0 4286  {csn 4590   cuni 4869   class class class wbr 5109  Refcref 22876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-ref 22879
This theorem is referenced by:  locfinref  32486
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