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Theorem naddcom 33601
Description: Natural addition commutes. (Contributed by Scott Fenton, 26-Aug-2024.)
Assertion
Ref Expression
naddcom ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))

Proof of Theorem naddcom
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7239 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
2 oveq2 7240 . . 3 (𝑎 = 𝑐 → (𝑏 +no 𝑎) = (𝑏 +no 𝑐))
31, 2eqeq12d 2754 . 2 (𝑎 = 𝑐 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝑐 +no 𝑏) = (𝑏 +no 𝑐)))
4 oveq2 7240 . . 3 (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
5 oveq1 7239 . . 3 (𝑏 = 𝑑 → (𝑏 +no 𝑐) = (𝑑 +no 𝑐))
64, 5eqeq12d 2754 . 2 (𝑏 = 𝑑 → ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
7 oveq1 7239 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
8 oveq2 7240 . . 3 (𝑎 = 𝑐 → (𝑑 +no 𝑎) = (𝑑 +no 𝑐))
97, 8eqeq12d 2754 . 2 (𝑎 = 𝑐 → ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
10 oveq1 7239 . . 3 (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
11 oveq2 7240 . . 3 (𝑎 = 𝐴 → (𝑏 +no 𝑎) = (𝑏 +no 𝐴))
1210, 11eqeq12d 2754 . 2 (𝑎 = 𝐴 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝐴 +no 𝑏) = (𝑏 +no 𝐴)))
13 oveq2 7240 . . 3 (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
14 oveq1 7239 . . 3 (𝑏 = 𝐵 → (𝑏 +no 𝐴) = (𝐵 +no 𝐴))
1513, 14eqeq12d 2754 . 2 (𝑏 = 𝐵 → ((𝐴 +no 𝑏) = (𝑏 +no 𝐴) ↔ (𝐴 +no 𝐵) = (𝐵 +no 𝐴)))
16 eleq1 2826 . . . . . . . . . . . 12 ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
1716ralimi 3085 . . . . . . . . . . 11 (∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
18 ralbi 3091 . . . . . . . . . . 11 (∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥) → (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
1917, 18syl 17 . . . . . . . . . 10 (∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
20193ad2ant3 1137 . . . . . . . . 9 ((∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
2120adantl 485 . . . . . . . 8 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
22 eleq1 2826 . . . . . . . . . . . 12 ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
2322ralimi 3085 . . . . . . . . . . 11 (∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
24 ralbi 3091 . . . . . . . . . . 11 (∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥) → (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
2523, 24syl 17 . . . . . . . . . 10 (∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
26253ad2ant2 1136 . . . . . . . . 9 ((∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
2726adantl 485 . . . . . . . 8 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
2821, 27anbi12d 634 . . . . . . 7 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ((∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥) ↔ (∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥)))
2928biancomd 467 . . . . . 6 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ((∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥) ↔ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)))
3029rabbidv 3403 . . . . 5 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → {𝑥 ∈ On ∣ (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥)} = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3130inteqd 4879 . . . 4 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → {𝑥 ∈ On ∣ (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥)} = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
32 naddov2 33600 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥)})
3332adantr 484 . . . 4 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥)})
34 naddov2 33600 . . . . . 6 ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑏 +no 𝑎) = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3534ancoms 462 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +no 𝑎) = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3635adantr 484 . . . 4 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑏 +no 𝑎) = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3731, 33, 363eqtr4d 2788 . . 3 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑎 +no 𝑏) = (𝑏 +no 𝑎))
3837ex 416 . 2 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (𝑎 +no 𝑏) = (𝑏 +no 𝑎)))
393, 6, 9, 12, 15, 38on2ind 33594 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2111  wral 3062  {crab 3066   cint 4874  Oncon0 6231  (class class class)co 7232   +no cnadd 33590
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 5194  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542
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 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-int 4875  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-tr 5177  df-id 5470  df-eprel 5475  df-po 5483  df-so 5484  df-fr 5524  df-se 5525  df-we 5526  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-pred 6176  df-ord 6234  df-on 6235  df-suc 6237  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-ov 7235  df-oprab 7236  df-mpo 7237  df-1st 7780  df-2nd 7781  df-frecs 8044  df-nadd 33591
This theorem is referenced by:  naddel2  33606  naddss2  33608
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