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Theorem bnj1143 30835
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 4513 . . . 4 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
2 notnotb 304 . . . . . . . 8 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
3 neq0 3922 . . . . . . . 8 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
42, 3xchbinx 324 . . . . . . 7 (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥𝐴)
5 df-rex 2915 . . . . . . . . 9 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑥(𝑥𝐴𝑧𝐵))
6 exsimpl 1793 . . . . . . . . 9 (∃𝑥(𝑥𝐴𝑧𝐵) → ∃𝑥 𝑥𝐴)
75, 6sylbi 207 . . . . . . . 8 (∃𝑥𝐴 𝑧𝐵 → ∃𝑥 𝑥𝐴)
87con3i 150 . . . . . . 7 (¬ ∃𝑥 𝑥𝐴 → ¬ ∃𝑥𝐴 𝑧𝐵)
94, 8sylbi 207 . . . . . 6 (𝐴 = ∅ → ¬ ∃𝑥𝐴 𝑧𝐵)
109alrimiv 1853 . . . . 5 (𝐴 = ∅ → ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
11 notnotb 304 . . . . . . 7 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
12 neq0 3922 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ∃𝑧 𝑧 𝑥𝐴 𝐵)
131eqeq1i 2625 . . . . . . . . 9 ( 𝑥𝐴 𝐵 = ∅ ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
1413notbii 310 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
15 df-iun 4513 . . . . . . . . . 10 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
1615eleq2i 2691 . . . . . . . . 9 (𝑧 𝑥𝐴 𝐵𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1716exbii 1772 . . . . . . . 8 (∃𝑧 𝑧 𝑥𝐴 𝐵 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1812, 14, 173bitr3i 290 . . . . . . 7 (¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1911, 18xchbinx 324 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
20 alnex 1704 . . . . . 6 (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
21 abid 2608 . . . . . . . 8 (𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ∃𝑥𝐴 𝑧𝐵)
2221notbii 310 . . . . . . 7 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑥𝐴 𝑧𝐵)
2322albii 1745 . . . . . 6 (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
2419, 20, 233bitr2i 288 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
2510, 24sylibr 224 . . . 4 (𝐴 = ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
261, 25syl5eq 2666 . . 3 (𝐴 = ∅ → 𝑥𝐴 𝐵 = ∅)
27 0ss 3963 . . 3 ∅ ⊆ 𝐵
2826, 27syl6eqss 3647 . 2 (𝐴 = ∅ → 𝑥𝐴 𝐵𝐵)
29 iunconst 4520 . . 3 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
30 eqimss 3649 . . 3 ( 𝑥𝐴 𝐵 = 𝐵 𝑥𝐴 𝐵𝐵)
3129, 30syl 17 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵𝐵)
3228, 31pm2.61ine 2874 1 𝑥𝐴 𝐵𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384  wal 1479   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wne 2791  wrex 2910  wss 3567  c0 3907   ciun 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-in 3574  df-ss 3581  df-nul 3908  df-iun 4513
This theorem is referenced by:  bnj1146  30836  bnj1145  31035  bnj1136  31039
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