Theorem infunsdom1 10125

Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 10126 assumes additionally that 𝐴 is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Assertion

Ref	Expression
infunsdom1	⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)

Proof of Theorem infunsdom1

Step	Hyp	Ref	Expression
1		simprl 771	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐴 ≼ 𝐵)
2		domsdomtr 9043	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
3	1, 2	sylan 581	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
4		unfi2 9213	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
5	3, 4	sylancom 589	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
6		simpllr 776	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → ω ≼ 𝑋)
7		sdomdomtr 9041	. . 3 ⊢ (((𝐴 ∪ 𝐵) ≺ ω ∧ ω ≼ 𝑋) → (𝐴 ∪ 𝐵) ≺ 𝑋)
8	5, 6, 7	syl2anc 585	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ 𝑋)
9		omelon 9558	. . . . . 6 ⊢ ω ∈ On
10		onenon 9864	. . . . . 6 ⊢ (ω ∈ On → ω ∈ dom card)
11	9, 10	ax-mp 5	. . . . 5 ⊢ ω ∈ dom card
12		simpll 767	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝑋 ∈ dom card)
13		sdomdom 8920	. . . . . . 7 ⊢ (𝐵 ≺ 𝑋 → 𝐵 ≼ 𝑋)
14	13	ad2antll 730	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ≼ 𝑋)
15		numdom 9951	. . . . . 6 ⊢ ((𝑋 ∈ dom card ∧ 𝐵 ≼ 𝑋) → 𝐵 ∈ dom card)
16	12, 14, 15	syl2anc 585	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ∈ dom card)
17		domtri2 9904	. . . . 5 ⊢ ((ω ∈ dom card ∧ 𝐵 ∈ dom card) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
18	11, 16, 17	sylancr 588	. . . 4 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
19	18	biimpar 477	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐵 ≺ ω) → ω ≼ 𝐵)
20		uncom 4099	. . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
21	16	adantr 480	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ∈ dom card)
22		simpr 484	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → ω ≼ 𝐵)
23	1	adantr 480	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐴 ≼ 𝐵)
24		infunabs 10119	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵 ∧ 𝐴 ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
25	21, 22, 23, 24	syl3anc 1374	. . . . 5 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
26	20, 25	eqbrtrid 5121	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐴 ∪ 𝐵) ≈ 𝐵)
27		simplrr 778	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ≺ 𝑋)
28		ensdomtr 9044	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ≈ 𝐵 ∧ 𝐵 ≺ 𝑋) → (𝐴 ∪ 𝐵) ≺ 𝑋)
29	26, 27, 28	syl2anc 585	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐴 ∪ 𝐵) ≺ 𝑋)
30	19, 29	syldan 592	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ 𝑋)
31	8, 30	pm2.61dan 813	1 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   ∪ cun 3888   class class class wbr 5086  dom cdm 5624  Oncon0 6317  ωcom 7810   ≈ cen 8883   ≼ cdom 8884   ≺ csdm 8885  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-oi 9418  df-dju 9816  df-card 9854

This theorem is referenced by:  infunsdom  10126