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Theorem naddov2 8626
Description: Alternate expression for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
Assertion
Ref Expression
naddov2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = {𝑥 ∈ On ∣ (∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑧,𝐴   𝑦,𝐵   𝑧,𝐵   𝑥,𝑦   𝑥,𝑧

Proof of Theorem naddov2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 naddov 8625 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
2 snssi 4769 . . . . . . . . 9 (𝐴 ∈ On → {𝐴} ⊆ On)
3 onss 7720 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
4 xpss12 5649 . . . . . . . . 9 (({𝐴} ⊆ On ∧ 𝐵 ⊆ On) → ({𝐴} × 𝐵) ⊆ (On × On))
52, 3, 4syl2an 597 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ (On × On))
6 naddfn 8622 . . . . . . . . 9 +no Fn (On × On)
76fndmi 6607 . . . . . . . 8 dom +no = (On × On)
85, 7sseqtrrdi 3996 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ dom +no )
9 fnfun 6603 . . . . . . . . 9 ( +no Fn (On × On) → Fun +no )
106, 9ax-mp 5 . . . . . . . 8 Fun +no
11 funimassov 7532 . . . . . . . 8 ((Fun +no ∧ ({𝐴} × 𝐵) ⊆ dom +no ) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑡 ∈ {𝐴}∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥))
1210, 11mpan 689 . . . . . . 7 (({𝐴} × 𝐵) ⊆ dom +no → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑡 ∈ {𝐴}∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥))
138, 12syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑡 ∈ {𝐴}∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥))
14 oveq1 7365 . . . . . . . . . 10 (𝑡 = 𝐴 → (𝑡 +no 𝑦) = (𝐴 +no 𝑦))
1514eleq1d 2823 . . . . . . . . 9 (𝑡 = 𝐴 → ((𝑡 +no 𝑦) ∈ 𝑥 ↔ (𝐴 +no 𝑦) ∈ 𝑥))
1615ralbidv 3175 . . . . . . . 8 (𝑡 = 𝐴 → (∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
1716ralsng 4635 . . . . . . 7 (𝐴 ∈ On → (∀𝑡 ∈ {𝐴}∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
1817adantr 482 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑡 ∈ {𝐴}∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
1913, 18bitrd 279 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
20 onss 7720 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ On)
21 snssi 4769 . . . . . . . . 9 (𝐵 ∈ On → {𝐵} ⊆ On)
22 xpss12 5649 . . . . . . . . 9 ((𝐴 ⊆ On ∧ {𝐵} ⊆ On) → (𝐴 × {𝐵}) ⊆ (On × On))
2320, 21, 22syl2an 597 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ (On × On))
2423, 7sseqtrrdi 3996 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ dom +no )
25 funimassov 7532 . . . . . . . 8 ((Fun +no ∧ (𝐴 × {𝐵}) ⊆ dom +no ) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧𝐴𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥))
2610, 25mpan 689 . . . . . . 7 ((𝐴 × {𝐵}) ⊆ dom +no → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧𝐴𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥))
2724, 26syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧𝐴𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥))
28 oveq2 7366 . . . . . . . . . 10 (𝑡 = 𝐵 → (𝑧 +no 𝑡) = (𝑧 +no 𝐵))
2928eleq1d 2823 . . . . . . . . 9 (𝑡 = 𝐵 → ((𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
3029ralsng 4635 . . . . . . . 8 (𝐵 ∈ On → (∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
3130ralbidv 3175 . . . . . . 7 (𝐵 ∈ On → (∀𝑧𝐴𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
3231adantl 483 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑧𝐴𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
3327, 32bitrd 279 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
3419, 33anbi12d 632 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥) ↔ (∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥)))
3534rabbidv 3416 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
3635inteqd 4913 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
371, 36eqtrd 2777 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = {𝑥 ∈ On ∣ (∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  {crab 3408  wss 3911  {csn 4587   cint 4908   × cxp 5632  dom cdm 5634  cima 5637  Oncon0 6318  Fun wfun 6491   Fn wfn 6492  (class class class)co 7358   +no cnadd 8612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-frecs 8213  df-nadd 8613
This theorem is referenced by:  naddcom  8629  naddid1  8630  naddssim  8631  naddelim  8632
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