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Theorem naddov2 33584
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 33583 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
2 snssi 4730 . . . . . . . . 9 (𝐴 ∈ On → {𝐴} ⊆ On)
3 onss 7577 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
4 xpss12 5575 . . . . . . . . 9 (({𝐴} ⊆ On ∧ 𝐵 ⊆ On) → ({𝐴} × 𝐵) ⊆ (On × On))
52, 3, 4syl2an 599 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ (On × On))
6 naddfn 33580 . . . . . . . . 9 +no Fn (On × On)
76fndmi 6491 . . . . . . . 8 dom +no = (On × On)
85, 7sseqtrrdi 3961 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ dom +no )
9 fnfun 6488 . . . . . . . . 9 ( +no Fn (On × On) → Fun +no )
106, 9ax-mp 5 . . . . . . . 8 Fun +no
11 funimassov 7394 . . . . . . . 8 ((Fun +no ∧ ({𝐴} × 𝐵) ⊆ dom +no ) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑡 ∈ {𝐴}∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥))
1210, 11mpan 690 . . . . . . 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 7229 . . . . . . . . . 10 (𝑡 = 𝐴 → (𝑡 +no 𝑦) = (𝐴 +no 𝑦))
1514eleq1d 2823 . . . . . . . . 9 (𝑡 = 𝐴 → ((𝑡 +no 𝑦) ∈ 𝑥 ↔ (𝐴 +no 𝑦) ∈ 𝑥))
1615ralbidv 3119 . . . . . . . 8 (𝑡 = 𝐴 → (∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
1716ralsng 4598 . . . . . . 7 (𝐴 ∈ On → (∀𝑡 ∈ {𝐴}∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
1817adantr 484 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑡 ∈ {𝐴}∀𝑦𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
1913, 18bitrd 282 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
20 onss 7577 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ On)
21 snssi 4730 . . . . . . . . 9 (𝐵 ∈ On → {𝐵} ⊆ On)
22 xpss12 5575 . . . . . . . . 9 ((𝐴 ⊆ On ∧ {𝐵} ⊆ On) → (𝐴 × {𝐵}) ⊆ (On × On))
2320, 21, 22syl2an 599 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ (On × On))
2423, 7sseqtrrdi 3961 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ dom +no )
25 funimassov 7394 . . . . . . . 8 ((Fun +no ∧ (𝐴 × {𝐵}) ⊆ dom +no ) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧𝐴𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥))
2610, 25mpan 690 . . . . . . 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 7230 . . . . . . . . . 10 (𝑡 = 𝐵 → (𝑧 +no 𝑡) = (𝑧 +no 𝐵))
2928eleq1d 2823 . . . . . . . . 9 (𝑡 = 𝐵 → ((𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
3029ralsng 4598 . . . . . . . 8 (𝐵 ∈ On → (∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
3130ralbidv 3119 . . . . . . 7 (𝐵 ∈ On → (∀𝑧𝐴𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
3231adantl 485 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑧𝐴𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
3327, 32bitrd 282 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
3419, 33anbi12d 634 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥) ↔ (∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥)))
3534rabbidv 3397 . . 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 4873 . 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 2778 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 209  wa 399   = wceq 1543  wcel 2111  wral 3062  {crab 3066  wss 3875  {csn 4550   cint 4868   × cxp 5558  dom cdm 5560  cima 5563  Oncon0 6222  Fun wfun 6383   Fn wfn 6384  (class class class)co 7222   +no cnadd 33574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5188  ax-sep 5201  ax-nul 5208  ax-pow 5267  ax-pr 5331  ax-un 7532
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-pss 3894  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4829  df-int 4869  df-iun 4915  df-br 5063  df-opab 5125  df-mpt 5145  df-tr 5171  df-id 5464  df-eprel 5469  df-po 5477  df-so 5478  df-fr 5518  df-se 5519  df-we 5520  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-pred 6169  df-ord 6225  df-on 6226  df-suc 6228  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-f1 6394  df-fo 6395  df-f1o 6396  df-fv 6397  df-ov 7225  df-oprab 7226  df-mpo 7227  df-1st 7770  df-2nd 7771  df-frecs 8032  df-nadd 33575
This theorem is referenced by:  naddcom  33585  naddid1  33586  naddssim  33587  naddelim  33588
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