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Theorem bnj1143 32303
 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 4888 . . . 4 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
2 notnotb 318 . . . . . . . 8 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
3 neq0 4246 . . . . . . . 8 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
42, 3xchbinx 337 . . . . . . 7 (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥𝐴)
5 df-rex 3076 . . . . . . . . 9 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑥(𝑥𝐴𝑧𝐵))
6 exsimpl 1869 . . . . . . . . 9 (∃𝑥(𝑥𝐴𝑧𝐵) → ∃𝑥 𝑥𝐴)
75, 6sylbi 220 . . . . . . . 8 (∃𝑥𝐴 𝑧𝐵 → ∃𝑥 𝑥𝐴)
87con3i 157 . . . . . . 7 (¬ ∃𝑥 𝑥𝐴 → ¬ ∃𝑥𝐴 𝑧𝐵)
94, 8sylbi 220 . . . . . 6 (𝐴 = ∅ → ¬ ∃𝑥𝐴 𝑧𝐵)
109alrimiv 1928 . . . . 5 (𝐴 = ∅ → ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
11 notnotb 318 . . . . . . 7 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
12 neq0 4246 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ∃𝑧 𝑧 𝑥𝐴 𝐵)
131eqeq1i 2763 . . . . . . . . 9 ( 𝑥𝐴 𝐵 = ∅ ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
1413notbii 323 . . . . . . . 8 𝑥𝐴 𝐵 = ∅ ↔ ¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
15 df-iun 4888 . . . . . . . . . 10 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
1615eleq2i 2843 . . . . . . . . 9 (𝑧 𝑥𝐴 𝐵𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1716exbii 1849 . . . . . . . 8 (∃𝑧 𝑧 𝑥𝐴 𝐵 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1812, 14, 173bitr3i 304 . . . . . . 7 (¬ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
1911, 18xchbinx 337 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
20 alnex 1783 . . . . . 6 (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
21 abid 2739 . . . . . . . 8 (𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ∃𝑥𝐴 𝑧𝐵)
2221notbii 323 . . . . . . 7 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ¬ ∃𝑥𝐴 𝑧𝐵)
2322albii 1821 . . . . . 6 (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} ↔ ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
2419, 20, 233bitr2i 302 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅ ↔ ∀𝑧 ¬ ∃𝑥𝐴 𝑧𝐵)
2510, 24sylibr 237 . . . 4 (𝐴 = ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} = ∅)
261, 25syl5eq 2805 . . 3 (𝐴 = ∅ → 𝑥𝐴 𝐵 = ∅)
27 0ss 4295 . . 3 ∅ ⊆ 𝐵
2826, 27eqsstrdi 3948 . 2 (𝐴 = ∅ → 𝑥𝐴 𝐵𝐵)
29 iunconst 4895 . . 3 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
30 eqimss 3950 . . 3 ( 𝑥𝐴 𝐵 = 𝐵 𝑥𝐴 𝐵𝐵)
3129, 30syl 17 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵𝐵)
3228, 31pm2.61ine 3034 1 𝑥𝐴 𝐵𝐵
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735   ≠ wne 2951  ∃wrex 3071   ⊆ wss 3860  ∅c0 4227  ∪ ciun 4886 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-in 3867  df-ss 3877  df-nul 4228  df-iun 4888 This theorem is referenced by:  bnj1146  32304  bnj1145  32506  bnj1136  32510
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