Theorem etaslts2 28210

Description: A version of etaslts 28209 with fewer hypotheses but a weaker upper bound. (Contributed by Scott Fenton, 10-Dec-2021.)

Assertion

Ref	Expression
etaslts2	⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ (bday‘𝑥) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem etaslts2

Step	Hyp	Ref	Expression
1		bdayfun 28163	. . . . . 6 ⊢ Fun bday
2		sltsex1 28179	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3		sltsex2 28180	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4		unexg 7784	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
5	2, 3, 4	syl2anc 596	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
6		funimaexg 6648	. . . . . 6 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → (bday “ (𝐴 ∪ 𝐵)) ∈ V)
7	1, 5, 6	sylancr 599	. . . . 5 ⊢ (𝐴 <<s 𝐵 → (bday “ (𝐴 ∪ 𝐵)) ∈ V)
8	7	uniexd 7783	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V)
9		imassrn 6210	. . . . . . 7 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
10		bdayrn 28167	. . . . . . 7 ⊢ ran bday = On
11	9, 10	sseqtri 3961	. . . . . 6 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ On
12		ssorduni 7817	. . . . . 6 ⊢ ((bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ (bday “ (𝐴 ∪ 𝐵)))
13	11, 12	ax-mp 5	. . . . 5 ⊢ Ord ∪ (bday “ (𝐴 ∪ 𝐵))
14		elon2 6391	. . . . 5 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ (Ord ∪ (bday “ (𝐴 ∪ 𝐵)) ∧ ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V))
15	13, 14	mpbiran 722	. . . 4 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V)
16	8, 15	sylibr 237	. . 3 ⊢ (𝐴 <<s 𝐵 → ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On)
17		onsucb 7852	. . 3 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On)
18	16, 17	sylib 221	. 2 ⊢ (𝐴 <<s 𝐵 → suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On)
19		onsucuni 7863	. . 3 ⊢ ((bday “ (𝐴 ∪ 𝐵)) ⊆ On → (bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
20	11, 19	mp1i 14	. 2 ⊢ (𝐴 <<s 𝐵 → (bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
21		etaslts 28209	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ∧ (bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵))) → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ (bday‘𝑥) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵))))
22	18, 20, 21	mpd3an23 1493	1 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ (bday‘𝑥) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵))))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  {csn 4588  ∪ cuni 4870   class class class wbr 5106  ran crn 5656   “ cima 5658  Ord word 6379  Oncon0 6380  suc csuc 6382  Fun wfun 6552  ‘cfv 6558  Nocsur 28027  bdaycbday 28029   <<s cslts 28173

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  df-slts 28174

This theorem is used by:  cutbdaybnd2  28212