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Theorem refun0 21223
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 2626 . . . 4 𝐴 = 𝐴
2 eqid 2626 . . . 4 𝐵 = 𝐵
31, 2refbas 21218 . . 3 (𝐴Ref𝐵 𝐵 = 𝐴)
43adantr 481 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → 𝐵 = 𝐴)
5 elun 3736 . . . 4 (𝑥 ∈ (𝐴 ∪ {∅}) ↔ (𝑥𝐴𝑥 ∈ {∅}))
6 refssex 21219 . . . . . 6 ((𝐴Ref𝐵𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
76adantlr 750 . . . . 5 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
8 0ss 3949 . . . . . . . . 9 ∅ ⊆ 𝑦
98a1i 11 . . . . . . . 8 ((𝐴Ref𝐵𝑦𝐵) → ∅ ⊆ 𝑦)
109reximdva0 3914 . . . . . . 7 ((𝐴Ref𝐵𝐵 ≠ ∅) → ∃𝑦𝐵 ∅ ⊆ 𝑦)
1110adantr 481 . . . . . 6 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦𝐵 ∅ ⊆ 𝑦)
12 elsni 4170 . . . . . . . 8 (𝑥 ∈ {∅} → 𝑥 = ∅)
13 sseq1 3610 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ⊆ 𝑦))
1413rexbidv 3050 . . . . . . . 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 247 . . . . 5 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦𝐵 𝑥𝑦)
187, 17jaodan 825 . . . 4 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥 ∈ {∅})) → ∃𝑦𝐵 𝑥𝑦)
195, 18sylan2b 492 . . 3 (((𝐴Ref𝐵𝐵 ≠ ∅) ∧ 𝑥 ∈ (𝐴 ∪ {∅})) → ∃𝑦𝐵 𝑥𝑦)
2019ralrimiva 2965 . 2 ((𝐴Ref𝐵𝐵 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦𝐵 𝑥𝑦)
21 refrel 21216 . . . . . 6 Rel Ref
2221brrelexi 5123 . . . . 5 (𝐴Ref𝐵𝐴 ∈ V)
23 p0ex 4818 . . . . 5 {∅} ∈ V
24 unexg 6913 . . . . 5 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ∪ {∅}) ∈ V)
2522, 23, 24sylancl 693 . . . 4 (𝐴Ref𝐵 → (𝐴 ∪ {∅}) ∈ V)
26 uniun 4427 . . . . . 6 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
27 0ex 4755 . . . . . . . 8 ∅ ∈ V
2827unisn 4422 . . . . . . 7 {∅} = ∅
2928uneq2i 3747 . . . . . 6 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
30 un0 3944 . . . . . 6 ( 𝐴 ∪ ∅) = 𝐴
3126, 29, 303eqtrri 2653 . . . . 5 𝐴 = (𝐴 ∪ {∅})
3231, 2isref 21217 . . . 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 956 1 ((𝐴Ref𝐵𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  Vcvv 3191  cun 3558  wss 3560  c0 3896  {csn 4153   cuni 4407   class class class wbr 4618  Refcref 21210
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-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-ref 21213
This theorem is referenced by:  locfinref  29682
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