Theorem disjsnxp 45424

Description: The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Assertion

Ref	Expression
disjsnxp	⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)

Distinct variable group:   𝐴,𝑗

Allowed substitution hint:   𝐵(𝑗)

Proof of Theorem disjsnxp

Step	Hyp	Ref	Expression
1		sndisj 5092	. . . 4 ⊢ Disj 𝑗 ∈ 𝐴 {𝑗}
2	1	a1i 11	. . 3 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 {𝑗})
3	2	disjxp1 45423	. 2 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
4	3	mptru 1549	1 ⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)

Syntax hints:  ⊤wtru 1543  {csn 4582  Disj wdisj 5067   × cxp 5630

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-disj 5068  df-opab 5163  df-xp 5638  df-rel 5639

This theorem is referenced by:  sge0xp  46781