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Theorem bnj1143 30987
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 4554 . . . 4 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
2 notnotb 304 . . . . . . . 8 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
3 neq0 3963 . . . . . . . 8 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
42, 3xchbinx 323 . . . . . . 7 (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥𝐴)
5 df-rex 2947 . . . . . . . . 9 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑥(𝑥𝐴𝑧𝐵))
6 exsimpl 1835 . . . . . . . . 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 1895 . . . . 5 (𝐴 = ∅ → ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
11 notnotb 304 . . . . . . 7 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
12 neq0 3963 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ∃𝑧 𝑧 𝑥𝐴 𝐵)
131eqeq1i 2656 . . . . . . . . 9 ( 𝑥𝐴 𝐵 = ∅ ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
1413notbii 309 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
15 df-iun 4554 . . . . . . . . . 10 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
1615eleq2i 2722 . . . . . . . . 9 (𝑧 𝑥𝐴 𝐵𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1716exbii 1814 . . . . . . . 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 323 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
20 alnex 1746 . . . . . 6 (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
21 abid 2639 . . . . . . . 8 (𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ∃𝑥𝐴 𝑧𝐵)
2221notbii 309 . . . . . . 7 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑥𝐴 𝑧𝐵)
2322albii 1787 . . . . . 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 2697 . . 3 (𝐴 = ∅ → 𝑥𝐴 𝐵 = ∅)
27 0ss 4005 . . 3 ∅ ⊆ 𝐵
2826, 27syl6eqss 3688 . 2 (𝐴 = ∅ → 𝑥𝐴 𝐵𝐵)
29 iunconst 4561 . . 3 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
30 eqimss 3690 . . 3 ( 𝑥𝐴 𝐵 = 𝐵 𝑥𝐴 𝐵𝐵)
3129, 30syl 17 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵𝐵)
3228, 31pm2.61ine 2906 1 𝑥𝐴 𝐵𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wrex 2942  wss 3607  c0 3948   ciun 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-iun 4554
This theorem is referenced by:  bnj1146  30988  bnj1145  31187  bnj1136  31191
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