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Theorem naddss1 8635
Description: Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
Assertion
Ref Expression
naddss1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))

Proof of Theorem naddss1
StepHypRef Expression
1 naddel1 8633 . . . 4 ((𝐵 ∈ On ∧ 𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐴 ↔ (𝐵 +no 𝐶) ∈ (𝐴 +no 𝐶)))
213com12 1124 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐴 ↔ (𝐵 +no 𝐶) ∈ (𝐴 +no 𝐶)))
32notbid 318 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ 𝐵𝐴 ↔ ¬ (𝐵 +no 𝐶) ∈ (𝐴 +no 𝐶)))
4 ontri1 6352 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
543adant3 1133 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
6 naddcl 8624 . . . 4 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +no 𝐶) ∈ On)
763adant2 1132 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +no 𝐶) ∈ On)
8 naddcl 8624 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) ∈ On)
983adant1 1131 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) ∈ On)
10 ontri1 6352 . . 3 (((𝐴 +no 𝐶) ∈ On ∧ (𝐵 +no 𝐶) ∈ On) → ((𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶) ↔ ¬ (𝐵 +no 𝐶) ∈ (𝐴 +no 𝐶)))
117, 9, 10syl2anc 585 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶) ↔ ¬ (𝐵 +no 𝐶) ∈ (𝐴 +no 𝐶)))
123, 5, 113bitr4d 311 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  w3a 1088  wcel 2107  wss 3911  Oncon0 6318  (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:  naddss2  8636  naddunif  8638
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