Theorem etaslts2 28185

Description: A version of etaslts 28184 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 28138	. . . . . 6 ⊢ Fun bday
2		sltsex1 28154	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3		sltsex2 28155	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4		unexg 7760	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
5	2, 3, 4	syl2anc 596	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
6		funimaexg 6626	. . . . . 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 7759	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V)
9		imassrn 6197	. . . . . . 7 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
10		bdayrn 28142	. . . . . . 7 ⊢ ran bday = On
11	9, 10	sseqtri 3956	. . . . . 6 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ On
12		ssorduni 7793	. . . . . 6 ⊢ ((bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ (bday “ (𝐴 ∪ 𝐵)))
13	11, 12	ax-mp 5	. . . . 5 ⊢ Ord ∪ (bday “ (𝐴 ∪ 𝐵))
14		elon2 6373	. . . . 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 7828	. . 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 7839	. . 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 28184	. 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 1492	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 2144  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  {csn 4584  ∪ cuni 4866   class class class wbr 5102  ran crn 5651   “ cima 5653  Ord word 6361  Oncon0 6362  suc csuc 6364  Fun wfun 6532  ‘cfv 6538  Nocsur 28002  bdaycbday 28004   <<s cslts 28148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149

This theorem is used by:  cutbdaybnd2  28187