Theorem nodenselem3 31082

Description: Lemma for nodense 31088. If one surreal is older than another, then there is an ordinal at which they are not equal. (Contributed by Scott Fenton, 16-Jun-2011.)

Assertion

Ref	Expression
nodenselem3	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( bday ‘𝐴) ∈ ( bday ‘𝐵) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))

Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem3

Step	Hyp	Ref	Expression
1		bdayval 31045	. . . 4 ⊢ (𝐵 ∈ No → ( bday ‘𝐵) = dom 𝐵)
2	1	adantl 481	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( bday ‘𝐵) = dom 𝐵)
3	2	eleq2d 2673	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( bday ‘𝐴) ∈ ( bday ‘𝐵) ↔ ( bday ‘𝐴) ∈ dom 𝐵))
4		bdayelon 31079	. . . 4 ⊢ ( bday ‘𝐴) ∈ On
5		nosgnn0 31055	. . . . . . . . 9 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜}
6		norn 31048	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ No → ran 𝐵 ⊆ {1𝑜, 2𝑜})
7	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ No ∧ ( bday ‘𝐴) ∈ dom 𝐵) → ran 𝐵 ⊆ {1𝑜, 2𝑜})
8		nofun 31046	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ No → Fun 𝐵)
9		fvelrn 6260	. . . . . . . . . . . 12 ⊢ ((Fun 𝐵 ∧ ( bday ‘𝐴) ∈ dom 𝐵) → (𝐵‘( bday ‘𝐴)) ∈ ran 𝐵)
10	8, 9	sylan 487	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ No ∧ ( bday ‘𝐴) ∈ dom 𝐵) → (𝐵‘( bday ‘𝐴)) ∈ ran 𝐵)
11	7, 10	sseldd 3569	. . . . . . . . . 10 ⊢ ((𝐵 ∈ No ∧ ( bday ‘𝐴) ∈ dom 𝐵) → (𝐵‘( bday ‘𝐴)) ∈ {1𝑜, 2𝑜})
12		eleq1 2676	. . . . . . . . . 10 ⊢ ((𝐵‘( bday ‘𝐴)) = ∅ → ((𝐵‘( bday ‘𝐴)) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
13	11, 12	syl5ibcom 234	. . . . . . . . 9 ⊢ ((𝐵 ∈ No ∧ ( bday ‘𝐴) ∈ dom 𝐵) → ((𝐵‘( bday ‘𝐴)) = ∅ → ∅ ∈ {1𝑜, 2𝑜}))
14	5, 13	mtoi 189	. . . . . . . 8 ⊢ ((𝐵 ∈ No ∧ ( bday ‘𝐴) ∈ dom 𝐵) → ¬ (𝐵‘( bday ‘𝐴)) = ∅)
15	14	neqned 2789	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ ( bday ‘𝐴) ∈ dom 𝐵) → (𝐵‘( bday ‘𝐴)) ≠ ∅)
16	15	adantll 746	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐴) ∈ dom 𝐵) → (𝐵‘( bday ‘𝐴)) ≠ ∅)
17		fvnobday 31081	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) = ∅)
18	17	ad2antrr 758	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐴) ∈ dom 𝐵) → (𝐴‘( bday ‘𝐴)) = ∅)
19	16, 18	neeqtrrd 2856	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐴) ∈ dom 𝐵) → (𝐵‘( bday ‘𝐴)) ≠ (𝐴‘( bday ‘𝐴)))
20	19	necomd 2837	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐴) ∈ dom 𝐵) → (𝐴‘( bday ‘𝐴)) ≠ (𝐵‘( bday ‘𝐴)))
21		fveq2 6103	. . . . . 6 ⊢ (𝑎 = ( bday ‘𝐴) → (𝐴‘𝑎) = (𝐴‘( bday ‘𝐴)))
22		fveq2 6103	. . . . . 6 ⊢ (𝑎 = ( bday ‘𝐴) → (𝐵‘𝑎) = (𝐵‘( bday ‘𝐴)))
23	21, 22	neeq12d 2843	. . . . 5 ⊢ (𝑎 = ( bday ‘𝐴) → ((𝐴‘𝑎) ≠ (𝐵‘𝑎) ↔ (𝐴‘( bday ‘𝐴)) ≠ (𝐵‘( bday ‘𝐴))))
24	23	rspcev 3282	. . . 4 ⊢ ((( bday ‘𝐴) ∈ On ∧ (𝐴‘( bday ‘𝐴)) ≠ (𝐵‘( bday ‘𝐴))) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎))
25	4, 20, 24	sylancr 694	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐴) ∈ dom 𝐵) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎))
26	25	ex 449	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( bday ‘𝐴) ∈ dom 𝐵 → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
27	3, 26	sylbid 229	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( bday ‘𝐴) ∈ ( bday ‘𝐵) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  {cpr 4127  dom cdm 5038  ran crn 5039  Oncon0 5640  Fun wfun 5798  ‘cfv 5804  1_𝑜c1o 7440  2_𝑜c2o 7441   No csur 31037   bday cbday 31039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-2o 7448  df-no 31040  df-bday 31042

This theorem is referenced by:  nodenselem4  31083