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Theorem disjxp1 6173
 Description: The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypothesis
Ref Expression
disjxp1.1 (𝜑Disj 𝑥𝐴 𝐵)
Assertion
Ref Expression
disjxp1 (𝜑Disj 𝑥𝐴 (𝐵 × 𝐶))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem disjxp1
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 6103 . . . . . . 7 (𝑦 ∈ (𝐵 × 𝐶) → (1st𝑦) ∈ 𝐵)
2 xp1st 6103 . . . . . . 7 (𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶) → (1st𝑦) ∈ 𝑤 / 𝑥𝐵)
3 disjxp1.1 . . . . . . . . . . . 12 (𝜑Disj 𝑥𝐴 𝐵)
4 df-disj 3939 . . . . . . . . . . . 12 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
53, 4sylib 121 . . . . . . . . . . 11 (𝜑 → ∀𝑧∃*𝑥𝐴 𝑧𝐵)
6 1stexg 6105 . . . . . . . . . . . . 13 (𝑦 ∈ V → (1st𝑦) ∈ V)
76elv 2713 . . . . . . . . . . . 12 (1st𝑦) ∈ V
8 eleq1 2217 . . . . . . . . . . . . 13 (𝑧 = (1st𝑦) → (𝑧𝐵 ↔ (1st𝑦) ∈ 𝐵))
98rmobidv 2642 . . . . . . . . . . . 12 (𝑧 = (1st𝑦) → (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑥𝐴 (1st𝑦) ∈ 𝐵))
107, 9spcv 2803 . . . . . . . . . . 11 (∀𝑧∃*𝑥𝐴 𝑧𝐵 → ∃*𝑥𝐴 (1st𝑦) ∈ 𝐵)
115, 10syl 14 . . . . . . . . . 10 (𝜑 → ∃*𝑥𝐴 (1st𝑦) ∈ 𝐵)
12 nfcv 2296 . . . . . . . . . . 11 𝑥𝐴
13 nfcv 2296 . . . . . . . . . . 11 𝑤𝐴
14 nfcsb1v 3060 . . . . . . . . . . . 12 𝑥𝑤 / 𝑥𝐵
1514nfel2 2309 . . . . . . . . . . 11 𝑥(1st𝑦) ∈ 𝑤 / 𝑥𝐵
16 csbeq1a 3036 . . . . . . . . . . . 12 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
1716eleq2d 2224 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((1st𝑦) ∈ 𝐵 ↔ (1st𝑦) ∈ 𝑤 / 𝑥𝐵))
1812, 13, 15, 17rmo4f 2906 . . . . . . . . . 10 (∃*𝑥𝐴 (1st𝑦) ∈ 𝐵 ↔ ∀𝑥𝐴𝑤𝐴 (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
1911, 18sylib 121 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴𝑤𝐴 (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
2019r19.21bi 2542 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑤𝐴 (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
2120r19.21bi 2542 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑤𝐴) → (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
221, 2, 21syl2ani 406 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑤𝐴) → ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
2322ralrimiva 2527 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑤𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
2423ralrimiva 2527 . . . 4 (𝜑 → ∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
25 nfcsb1v 3060 . . . . . . 7 𝑥𝑤 / 𝑥𝐶
2614, 25nfxp 4606 . . . . . 6 𝑥(𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)
2726nfel2 2309 . . . . 5 𝑥 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)
28 csbeq1a 3036 . . . . . . 7 (𝑥 = 𝑤𝐶 = 𝑤 / 𝑥𝐶)
2916, 28xpeq12d 4604 . . . . . 6 (𝑥 = 𝑤 → (𝐵 × 𝐶) = (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶))
3029eleq2d 2224 . . . . 5 (𝑥 = 𝑤 → (𝑦 ∈ (𝐵 × 𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)))
3112, 13, 27, 30rmo4f 2906 . . . 4 (∃*𝑥𝐴 𝑦 ∈ (𝐵 × 𝐶) ↔ ∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
3224, 31sylibr 133 . . 3 (𝜑 → ∃*𝑥𝐴 𝑦 ∈ (𝐵 × 𝐶))
3332alrimiv 1851 . 2 (𝜑 → ∀𝑦∃*𝑥𝐴 𝑦 ∈ (𝐵 × 𝐶))
34 df-disj 3939 . 2 (Disj 𝑥𝐴 (𝐵 × 𝐶) ↔ ∀𝑦∃*𝑥𝐴 𝑦 ∈ (𝐵 × 𝐶))
3533, 34sylibr 133 1 (𝜑Disj 𝑥𝐴 (𝐵 × 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332   ∈ wcel 2125  ∀wral 2432  ∃*wrmo 2435  Vcvv 2709  ⦋csb 3027  Disj wdisj 3938   × cxp 4577  ‘cfv 5163  1st c1st 6076 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rmo 2440  df-v 2711  df-sbc 2934  df-csb 3028  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-disj 3939  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-fo 5169  df-fv 5171  df-1st 6078 This theorem is referenced by:  disjsnxp  6174
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