Theorem xpwf 45314

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 9734	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
2		pwwf 9731	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
3		pwwf 9731	. . 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 5766	. . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)
6		sswf 9732	. . 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 3901   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865   × cxp 5630   “ cima 5635  Oncon0 6325  𝑅₁cr1 9686

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 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-r1 9688  df-rank 9689

This theorem is referenced by:  relwf  45317