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Theorem naddcom 8680
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 7411 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
2 oveq2 7412 . . 3 (𝑎 = 𝑐 → (𝑏 +no 𝑎) = (𝑏 +no 𝑐))
31, 2eqeq12d 2742 . 2 (𝑎 = 𝑐 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝑐 +no 𝑏) = (𝑏 +no 𝑐)))
4 oveq2 7412 . . 3 (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
5 oveq1 7411 . . 3 (𝑏 = 𝑑 → (𝑏 +no 𝑐) = (𝑑 +no 𝑐))
64, 5eqeq12d 2742 . 2 (𝑏 = 𝑑 → ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
7 oveq1 7411 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
8 oveq2 7412 . . 3 (𝑎 = 𝑐 → (𝑑 +no 𝑎) = (𝑑 +no 𝑐))
97, 8eqeq12d 2742 . 2 (𝑎 = 𝑐 → ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
10 oveq1 7411 . . 3 (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
11 oveq2 7412 . . 3 (𝑎 = 𝐴 → (𝑏 +no 𝑎) = (𝑏 +no 𝐴))
1210, 11eqeq12d 2742 . 2 (𝑎 = 𝐴 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝐴 +no 𝑏) = (𝑏 +no 𝐴)))
13 oveq2 7412 . . 3 (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
14 oveq1 7411 . . 3 (𝑏 = 𝐵 → (𝑏 +no 𝐴) = (𝐵 +no 𝐴))
1513, 14eqeq12d 2742 . 2 (𝑏 = 𝐵 → ((𝐴 +no 𝑏) = (𝑏 +no 𝐴) ↔ (𝐴 +no 𝐵) = (𝐵 +no 𝐴)))
16 eleq1 2815 . . . . . . . . . . . 12 ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
1716ralimi 3077 . . . . . . . . . . 11 (∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
18 ralbi 3097 . . . . . . . . . . 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 1132 . . . . . . . . 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 2815 . . . . . . . . . . . 12 ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
2322ralimi 3077 . . . . . . . . . . 11 (∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
24 ralbi 3097 . . . . . . . . . . 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 1131 . . . . . . . . 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 3434 . . . . 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 4948 . . . 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 8677 . . . . 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 8677 . . . . . 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 2776 . . 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 8667 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 1084   = wceq 1533  wcel 2098  wral 3055  {crab 3426   cint 4943  Oncon0 6357  (class class class)co 7404   +no cnadd 8663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-frecs 8264  df-nadd 8664
This theorem is referenced by:  naddlid  8682  naddel2  8686  naddss2  8688  naddword2  8690  nadd32  8695  nadd42  8697  addsproplem2  27837  nadd2rabex  42694  nadd1rabtr  42696  nadd1rabex  42698  naddwordnexlem4  42710
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