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Theorem disjxp1 6231
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 6160 . . . . . . 7 (𝑦 ∈ (𝐵 × 𝐶) → (1st𝑦) ∈ 𝐵)
2 xp1st 6160 . . . . . . 7 (𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶) → (1st𝑦) ∈ 𝑤 / 𝑥𝐵)
3 disjxp1.1 . . . . . . . . . . . 12 (𝜑Disj 𝑥𝐴 𝐵)
4 df-disj 3978 . . . . . . . . . . . 12 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
53, 4sylib 122 . . . . . . . . . . 11 (𝜑 → ∀𝑧∃*𝑥𝐴 𝑧𝐵)
6 1stexg 6162 . . . . . . . . . . . . 13 (𝑦 ∈ V → (1st𝑦) ∈ V)
76elv 2741 . . . . . . . . . . . 12 (1st𝑦) ∈ V
8 eleq1 2240 . . . . . . . . . . . . 13 (𝑧 = (1st𝑦) → (𝑧𝐵 ↔ (1st𝑦) ∈ 𝐵))
98rmobidv 2665 . . . . . . . . . . . 12 (𝑧 = (1st𝑦) → (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑥𝐴 (1st𝑦) ∈ 𝐵))
107, 9spcv 2831 . . . . . . . . . . 11 (∀𝑧∃*𝑥𝐴 𝑧𝐵 → ∃*𝑥𝐴 (1st𝑦) ∈ 𝐵)
115, 10syl 14 . . . . . . . . . 10 (𝜑 → ∃*𝑥𝐴 (1st𝑦) ∈ 𝐵)
12 nfcv 2319 . . . . . . . . . . 11 𝑥𝐴
13 nfcv 2319 . . . . . . . . . . 11 𝑤𝐴
14 nfcsb1v 3090 . . . . . . . . . . . 12 𝑥𝑤 / 𝑥𝐵
1514nfel2 2332 . . . . . . . . . . 11 𝑥(1st𝑦) ∈ 𝑤 / 𝑥𝐵
16 csbeq1a 3066 . . . . . . . . . . . 12 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
1716eleq2d 2247 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((1st𝑦) ∈ 𝐵 ↔ (1st𝑦) ∈ 𝑤 / 𝑥𝐵))
1812, 13, 15, 17rmo4f 2935 . . . . . . . . . 10 (∃*𝑥𝐴 (1st𝑦) ∈ 𝐵 ↔ ∀𝑥𝐴𝑤𝐴 (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
1911, 18sylib 122 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴𝑤𝐴 (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
2019r19.21bi 2565 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑤𝐴 (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
2120r19.21bi 2565 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑤𝐴) → (((1st𝑦) ∈ 𝐵 ∧ (1st𝑦) ∈ 𝑤 / 𝑥𝐵) → 𝑥 = 𝑤))
221, 2, 21syl2ani 408 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑤𝐴) → ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
2322ralrimiva 2550 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑤𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
2423ralrimiva 2550 . . . 4 (𝜑 → ∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
25 nfcsb1v 3090 . . . . . . 7 𝑥𝑤 / 𝑥𝐶
2614, 25nfxp 4650 . . . . . 6 𝑥(𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)
2726nfel2 2332 . . . . 5 𝑥 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)
28 csbeq1a 3066 . . . . . . 7 (𝑥 = 𝑤𝐶 = 𝑤 / 𝑥𝐶)
2916, 28xpeq12d 4648 . . . . . 6 (𝑥 = 𝑤 → (𝐵 × 𝐶) = (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶))
3029eleq2d 2247 . . . . 5 (𝑥 = 𝑤 → (𝑦 ∈ (𝐵 × 𝐶) ↔ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)))
3112, 13, 27, 30rmo4f 2935 . . . 4 (∃*𝑥𝐴 𝑦 ∈ (𝐵 × 𝐶) ↔ ∀𝑥𝐴𝑤𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (𝑤 / 𝑥𝐵 × 𝑤 / 𝑥𝐶)) → 𝑥 = 𝑤))
3224, 31sylibr 134 . . 3 (𝜑 → ∃*𝑥𝐴 𝑦 ∈ (𝐵 × 𝐶))
3332alrimiv 1874 . 2 (𝜑 → ∀𝑦∃*𝑥𝐴 𝑦 ∈ (𝐵 × 𝐶))
34 df-disj 3978 . 2 (Disj 𝑥𝐴 (𝐵 × 𝐶) ↔ ∀𝑦∃*𝑥𝐴 𝑦 ∈ (𝐵 × 𝐶))
3533, 34sylibr 134 1 (𝜑Disj 𝑥𝐴 (𝐵 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351   = wceq 1353  wcel 2148  wral 2455  ∃*wrmo 2458  Vcvv 2737  csb 3057  Disj wdisj 3977   × cxp 4621  cfv 5212  1st c1st 6133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rmo 2463  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fo 5218  df-fv 5220  df-1st 6135
This theorem is referenced by:  disjsnxp  6232
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