Theorem xpwf 45409

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 5758	. . 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 692	. 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 2114   ∪ cun 3888   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851   × cxp 5622   “ cima 5627  Oncon0 6317  𝑅₁cr1 9677

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 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-r1 9679  df-rank 9680

This theorem is referenced by:  relwf  45412