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Theorem bnj1143 33442
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1143 𝑥𝐴 𝐵𝐵
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bnj1143
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iun 4961 . . . 4 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
2 notnotb 315 . . . . . . . 8 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
3 neq0 4310 . . . . . . . 8 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
42, 3xchbinx 334 . . . . . . 7 (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥𝐴)
5 df-rex 3075 . . . . . . . . 9 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑥(𝑥𝐴𝑧𝐵))
6 exsimpl 1872 . . . . . . . . 9 (∃𝑥(𝑥𝐴𝑧𝐵) → ∃𝑥 𝑥𝐴)
75, 6sylbi 216 . . . . . . . 8 (∃𝑥𝐴 𝑧𝐵 → ∃𝑥 𝑥𝐴)
87con3i 154 . . . . . . 7 (¬ ∃𝑥 𝑥𝐴 → ¬ ∃𝑥𝐴 𝑧𝐵)
94, 8sylbi 216 . . . . . 6 (𝐴 = ∅ → ¬ ∃𝑥𝐴 𝑧𝐵)
109alrimiv 1931 . . . . 5 (𝐴 = ∅ → ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
11 notnotb 315 . . . . . . 7 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
12 neq0 4310 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ∃𝑧 𝑧 𝑥𝐴 𝐵)
131eqeq1i 2742 . . . . . . . . 9 ( 𝑥𝐴 𝐵 = ∅ ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
1413notbii 320 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
15 df-iun 4961 . . . . . . . . . 10 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
1615eleq2i 2830 . . . . . . . . 9 (𝑧 𝑥𝐴 𝐵𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1716exbii 1851 . . . . . . . 8 (∃𝑧 𝑧 𝑥𝐴 𝐵 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1812, 14, 173bitr3i 301 . . . . . . 7 (¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1911, 18xchbinx 334 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
20 alnex 1784 . . . . . 6 (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
21 abid 2718 . . . . . . . 8 (𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ∃𝑥𝐴 𝑧𝐵)
2221notbii 320 . . . . . . 7 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑥𝐴 𝑧𝐵)
2322albii 1822 . . . . . 6 (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
2419, 20, 233bitr2i 299 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
2510, 24sylibr 233 . . . 4 (𝐴 = ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
261, 25eqtrid 2789 . . 3 (𝐴 = ∅ → 𝑥𝐴 𝐵 = ∅)
27 0ss 4361 . . 3 ∅ ⊆ 𝐵
2826, 27eqsstrdi 4003 . 2 (𝐴 = ∅ → 𝑥𝐴 𝐵𝐵)
29 iunconst 4968 . . 3 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
30 eqimss 4005 . . 3 ( 𝑥𝐴 𝐵 = 𝐵 𝑥𝐴 𝐵𝐵)
3129, 30syl 17 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵𝐵)
3228, 31pm2.61ine 3029 1 𝑥𝐴 𝐵𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2714  wne 2944  wrex 3074  wss 3915  c0 4287   ciun 4959
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-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-v 3450  df-dif 3918  df-in 3922  df-ss 3932  df-nul 4288  df-iun 4961
This theorem is referenced by:  bnj1146  33443  bnj1145  33645  bnj1136  33649
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