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Theorem disjxp1 6099
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 6029 . . . . . . 7 (𝑦 ∈ (𝐵 × 𝐶) → (1st𝑦) ∈ 𝐵)
2 xp1st 6029 . . . . . . 7 (𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶) → (1st𝑦) ∈ 𝑤 / 𝑥𝐵)
3 disjxp1.1 . . . . . . . . . . . 12 (𝜑Disj 𝑥𝐴 𝐵)
4 df-disj 3875 . . . . . . . . . . . 12 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
53, 4sylib 121 . . . . . . . . . . 11 (𝜑 → ∀𝑧∃*𝑥𝐴 𝑧𝐵)
6 1stexg 6031 . . . . . . . . . . . . 13 (𝑦 ∈ V → (1st𝑦) ∈ V)
76elv 2662 . . . . . . . . . . . 12 (1st𝑦) ∈ V
8 eleq1 2178 . . . . . . . . . . . . 13 (𝑧 = (1st𝑦) → (𝑧𝐵 ↔ (1st𝑦) ∈ 𝐵))
98rmobidv 2594 . . . . . . . . . . . 12 (𝑧 = (1st𝑦) → (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑥𝐴 (1st𝑦) ∈ 𝐵))
107, 9spcv 2751 . . . . . . . . . . 11 (∀𝑧∃*𝑥𝐴 𝑧𝐵 → ∃*𝑥𝐴 (1st𝑦) ∈ 𝐵)
115, 10syl 14 . . . . . . . . . 10 (𝜑 → ∃*𝑥𝐴 (1st𝑦) ∈ 𝐵)
12 nfcv 2256 . . . . . . . . . . 11 𝑥𝐴
13 nfcv 2256 . . . . . . . . . . 11 𝑤𝐴
14 nfcsb1v 3003 . . . . . . . . . . . 12 𝑥𝑤 / 𝑥𝐵
1514nfel2 2269 . . . . . . . . . . 11 𝑥(1st𝑦) ∈ 𝑤 / 𝑥𝐵
16 csbeq1a 2981 . . . . . . . . . . . 12 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
1716eleq2d 2185 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((1st𝑦) ∈ 𝐵 ↔ (1st𝑦) ∈ 𝑤 / 𝑥𝐵))
1812, 13, 15, 17rmo4f 2853 . . . . . . . . . 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 2495 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑤𝐴 (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
2120r19.21bi 2495 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑤𝐴) → (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
221, 2, 21syl2ani 403 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑤𝐴) → ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
2322ralrimiva 2480 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑤𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
2423ralrimiva 2480 . . . 4 (𝜑 → ∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
25 nfcsb1v 3003 . . . . . . 7 𝑥𝑤 / 𝑥𝐶
2614, 25nfxp 4534 . . . . . 6 𝑥(𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)
2726nfel2 2269 . . . . 5 𝑥 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)
28 csbeq1a 2981 . . . . . . 7 (𝑥 = 𝑤𝐶 = 𝑤 / 𝑥𝐶)
2916, 28xpeq12d 4532 . . . . . 6 (𝑥 = 𝑤 → (𝐵 × 𝐶) = (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶))
3029eleq2d 2185 . . . . 5 (𝑥 = 𝑤 → (𝑦 ∈ (𝐵 × 𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)))
3112, 13, 27, 30rmo4f 2853 . . . 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 1828 . 2 (𝜑 → ∀𝑦∃*𝑥𝐴 𝑦 ∈ (𝐵 × 𝐶))
34 df-disj 3875 . 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 1312   = wceq 1314  wcel 1463  wral 2391  ∃*wrmo 2394  Vcvv 2658  csb 2973  Disj wdisj 3874   × cxp 4505  cfv 5091  1st c1st 6002
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rmo 2399  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-disj 3875  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fo 5097  df-fv 5099  df-1st 6004
This theorem is referenced by:  disjsnxp  6100
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