Theorem nosepne 32343

Description: The value of two non-equal surreals at the first place they differ is different. (Contributed by Scott Fenton, 24-Nov-2021.)

Assertion

Ref	Expression
nosepne	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosepne

Step	Hyp	Ref	Expression
1		sltso 32339	. . . 4 ⊢ <s Or No
2		sotrine 32171	. . . 4 ⊢ (( <s Or No ∧ (𝐴 ∈ No ∧ 𝐵 ∈ No )) → (𝐴 ≠ 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐵 <s 𝐴)))
3	1, 2	mpan 682	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≠ 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐵 <s 𝐴)))
4		nosepnelem 32342	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))
5	4	3expia 1151	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})))
6		nosepnelem 32342	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ 𝐵 <s 𝐴) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}))
7		necom 3025	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ (𝐵‘𝑥) ≠ (𝐴‘𝑥))
8	7	rabbii 3370	. . . . . . . . . . 11 ⊢ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} = {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}
9	8	inteqi 4672	. . . . . . . . . 10 ⊢ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}
10	9	fveq2i 6415	. . . . . . . . 9 ⊢ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)})
11	9	fveq2i 6415	. . . . . . . . 9 ⊢ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)})
12	10, 11	neeq12i 3038	. . . . . . . 8 ⊢ ((𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ↔ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}))
13		necom 3025	. . . . . . . 8 ⊢ ((𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ↔ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}))
14	12, 13	bitri 267	. . . . . . 7 ⊢ ((𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ↔ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}))
15	6, 14	sylibr 226	. . . . . 6 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ 𝐵 <s 𝐴) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))
16	15	3expia 1151	. . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 <s 𝐴 → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})))
17	16	ancoms 451	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐵 <s 𝐴 → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})))
18	5, 17	jaod 886	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 <s 𝐵 ∨ 𝐵 <s 𝐴) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})))
19	3, 18	sylbid 232	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≠ 𝐵 → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})))
20	19	3impia 1146	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∨ wo 874   ∧ w3a 1108   ∈ wcel 2157   ≠ wne 2972  {crab 3094  ∩ cint 4668   class class class wbr 4844   Or wor 5233  Oncon0 5942  ‘cfv 6102   No csur 32305   <s cslt 32306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-ord 5945  df-on 5946  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110  df-1o 7800  df-2o 7801  df-no 32308  df-slt 32309

This theorem is referenced by:  nosep1o  32344  nosepssdm  32348  noresle  32358  noetalem3  32377