Theorem disjsnxp 45904

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 5095	. . . 4 ⊢ Disj 𝑗 ∈ 𝐴 {𝑗}
2	1	a1i 11	. . 3 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 {𝑗})
3	2	disjxp1 45903	. 2 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
4	3	mptru 1577	1 ⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)

This proof depends on syntax axioms:  ⊤wtru 1571  {csn 4584  Disj wdisj 5070   × cxp 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-disj 5071  df-opab 5168  df-xp 5661  df-rel 5662

This theorem is used by:  sge0xp  47257