Theorem opelxp1 4757

Description: The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opelxp1	⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴 ∈ 𝐶)

Proof of Theorem opelxp1

Step	Hyp	Ref	Expression
1		opelxp 4753	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
2	1	simplbi 274	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2200  ⟨cop 3670   × cxp 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-opab 4149  df-xp 4729

This theorem is referenced by:  otelxp1  4759  dmxpss  5165  nfvres  5671  ressnop0  5830  swoord1  6726  swoord2  6727  txlm  14993