Theorem etasslt2 27795

Description: A version of etasslt 27794 with fewer hypotheses but a weaker upper bound. (Contributed by Scott Fenton, 10-Dec-2021.)

Assertion

Ref	Expression
etasslt2	⊢ (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 etasslt2

Step	Hyp	Ref	Expression
1		bdayfun 27753	. . . . . 6 ⊢ Fun bday
2		ssltex1 27767	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3		ssltex2 27768	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4		unexg 7745	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
5	2, 3, 4	syl2anc 584	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
6		funimaexg 6633	. . . . . 6 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
7	1, 5, 6	sylancr 587	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
8	7	uniexd 7744	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
9		imassrn 6069	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
10		bdayrn 27756	. . . . . . 7 ⊢ ran bday = On
11	9, 10	sseqtri 4012	. . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On
12		ssorduni 7781	. . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵)))
13	11, 12	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵))
14		elon2 6374	. . . . 5 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ (Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) ∧ ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V))
15	13, 14	mpbiran 709	. . . 4 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
16	8, 15	sylibr 234	. . 3 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On)
17		onsucb 7819	. . 3 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On)
18	16, 17	sylib 218	. 2 ⊢ (𝐴 <<s 𝐵 → suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On)
19		onsucuni 7830	. . 3 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → ( bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
20	11, 19	mp1i 13	. 2 ⊢ (𝐴 <<s 𝐵 → ( bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
21		etasslt 27794	. 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 1464	1 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2107  ∃wrex 3059  Vcvv 3463   ∪ cun 3929   ⊆ wss 3931  {csn 4606  ∪ cuni 4887   class class class wbr 5123  ran crn 5666   “ cima 5668  Ord word 6362  Oncon0 6363  suc csuc 6365  Fun wfun 6535  ‘cfv 6541   No csur 27620   bday cbday 27622   <<s csslt 27761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-int 4927  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-1o 8488  df-2o 8489  df-no 27623  df-slt 27624  df-bday 27625  df-sslt 27762

This theorem is referenced by:  scutbdaybnd2  27797