Theorem xpwf 44961

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 9770	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
2		pwwf 9767	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
3		pwwf 9767	. . 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 5775	. . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)
6		sswf 9768	. . 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 3915   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874   × cxp 5639   “ cima 5644  Oncon0 6335  𝑅₁cr1 9722

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-r1 9724  df-rank 9725

This theorem is referenced by:  relwf  44964