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 9725	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
2		pwwf 9722	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
3		pwwf 9722	. . 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 5756	. . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)
6		sswf 9723	. . 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 2109   ∪ cun 3903   ⊆ wss 3905  𝒫 cpw 4553  ∪ cuni 4861   × cxp 5621   “ cima 5626  Oncon0 6311  𝑅₁cr1 9677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9679  df-rank 9680

This theorem is referenced by:  relwf  44941