Theorem xpwf 44938

Description: The Cartesian product of two well-founded sets is well-founded. (Contributed by Eric Schmidt, 12-Sep-2025.)

Assertion

Ref	Expression
xpwf	⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 × 𝐵) ∈ ∪ (𝑅1 “ On))

Proof of Theorem xpwf

Step	Hyp	Ref	Expression
1		unwf 9847	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
2		pwwf 9844	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
3		pwwf 9844	. . 3 ⊢ (𝒫 (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
4	1, 2, 3	3bitri 297	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
5		xpsspw 5821	. . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)
6		sswf 9845	. . 3 ⊢ ((𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ∧ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)) → (𝐴 × 𝐵) ∈ ∪ (𝑅1 “ On))
7	5, 6	mpan2 691	. 2 ⊢ (𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐴 × 𝐵) ∈ ∪ (𝑅1 “ On))
8	4, 7	sylbi 217	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 × 𝐵) ∈ ∪ (𝑅1 “ On))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105   ∪ cun 3960   ⊆ wss 3962  𝒫 cpw 4604  ∪ cuni 4911   × cxp 5686   “ cima 5691  Oncon0 6385  𝑅₁cr1 9799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-r1 9801  df-rank 9802

This theorem is referenced by:  relwf  44941