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Theorem naddcom 33762
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 7262 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
2 oveq2 7263 . . 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 7263 . . 3 (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
5 oveq1 7262 . . 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 7262 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
8 oveq2 7263 . . 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 7262 . . 3 (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
11 oveq2 7263 . . 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 7263 . . 3 (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
14 oveq1 7262 . . 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 3086 . . . . . . . . . . 11 (∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
18 ralbi 3092 . . . . . . . . . . 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 1133 . . . . . . . . 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 481 . . . . . . . 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 3086 . . . . . . . . . . 11 (∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
24 ralbi 3092 . . . . . . . . . . 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 1132 . . . . . . . . 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 481 . . . . . . . 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 630 . . . . . . 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 463 . . . . . 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 3404 . . . . 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 4881 . . . 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 33761 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥)})
3332adantr 480 . . . 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 33761 . . . . . 6 ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑏 +no 𝑎) = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3534ancoms 458 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +no 𝑎) = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3635adantr 480 . . . 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 412 . 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 33755 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1539  wcel 2108  wral 3063  {crab 3067   cint 4876  Oncon0 6251  (class class class)co 7255   +no cnadd 33751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-frecs 8068  df-nadd 33752
This theorem is referenced by:  naddel2  33767  naddss2  33769
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