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Theorem bnj1143 31179
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 4714 . . . 4 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
2 notnotb 306 . . . . . . . 8 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
3 neq0 4131 . . . . . . . 8 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
42, 3xchbinx 325 . . . . . . 7 (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥𝐴)
5 df-rex 3102 . . . . . . . . 9 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑥(𝑥𝐴𝑧𝐵))
6 exsimpl 1956 . . . . . . . . 9 (∃𝑥(𝑥𝐴𝑧𝐵) → ∃𝑥 𝑥𝐴)
75, 6sylbi 208 . . . . . . . 8 (∃𝑥𝐴 𝑧𝐵 → ∃𝑥 𝑥𝐴)
87con3i 151 . . . . . . 7 (¬ ∃𝑥 𝑥𝐴 → ¬ ∃𝑥𝐴 𝑧𝐵)
94, 8sylbi 208 . . . . . 6 (𝐴 = ∅ → ¬ ∃𝑥𝐴 𝑧𝐵)
109alrimiv 2018 . . . . 5 (𝐴 = ∅ → ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
11 notnotb 306 . . . . . . 7 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
12 neq0 4131 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ∃𝑧 𝑧 𝑥𝐴 𝐵)
131eqeq1i 2811 . . . . . . . . 9 ( 𝑥𝐴 𝐵 = ∅ ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
1413notbii 311 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
15 df-iun 4714 . . . . . . . . . 10 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
1615eleq2i 2877 . . . . . . . . 9 (𝑧 𝑥𝐴 𝐵𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1716exbii 1933 . . . . . . . 8 (∃𝑧 𝑧 𝑥𝐴 𝐵 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1812, 14, 173bitr3i 292 . . . . . . 7 (¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1911, 18xchbinx 325 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
20 alnex 1861 . . . . . 6 (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
21 abid 2794 . . . . . . . 8 (𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ∃𝑥𝐴 𝑧𝐵)
2221notbii 311 . . . . . . 7 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑥𝐴 𝑧𝐵)
2322albii 1904 . . . . . 6 (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
2419, 20, 233bitr2i 290 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
2510, 24sylibr 225 . . . 4 (𝐴 = ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
261, 25syl5eq 2852 . . 3 (𝐴 = ∅ → 𝑥𝐴 𝐵 = ∅)
27 0ss 4170 . . 3 ∅ ⊆ 𝐵
2826, 27syl6eqss 3852 . 2 (𝐴 = ∅ → 𝑥𝐴 𝐵𝐵)
29 iunconst 4721 . . 3 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
30 eqimss 3854 . . 3 ( 𝑥𝐴 𝐵 = 𝐵 𝑥𝐴 𝐵𝐵)
3129, 30syl 17 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵𝐵)
3228, 31pm2.61ine 3061 1 𝑥𝐴 𝐵𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wne 2978  wrex 3097  wss 3769  c0 4116   ciun 4712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-v 3393  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4117  df-iun 4714
This theorem is referenced by:  bnj1146  31180  bnj1145  31379  bnj1136  31383
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